Extremal Hypergraphs

Oleg Pikhurko

Department of Pure Mathematics and Mathematical Statistics

Cambridge University, UK and Trinity College, Cambridge, UK

         E-mail: O.Pikhurko@dpmms.cam.ac.uk

October 11, 1999

Declaration

This is to certify that this dissertation is the outcome of my own work and no part was done in collaboration. To the best of my knowledge, all presented here results are original and new unless explicitly stated otherwise. No part of this dissertation has been submitted for a degree or a diploma at any other institution.

I want to express my extreme gratitude to Andrew Thomason, my research supervisor, for the very stimulating discussions, comments and encouragement which were of great help in the work. Without him this thesis could not have been possible.

I am thankful to Trinity College, Cambridge, for the financial support by an External Research Studentship and for the other kinds of support.

Oleg Pikhurko

Cambridge

December 21, 1999

Introduction

In this thesis we consider different extremal problems for set systems. The extremal (hyper-)graph theory has long been regarded as an important subject comprising a large number of various problems and methods.

Of course, we do not even try to present here all the features of the theory. Instead, we consider a few different facets such as saturated hypergraphs, weakly saturated hypergraphs, minimum chain decompositions, enumeration results for hypertrees, and size Ramsey numbers. We try to demonstrate different proof techniques in action and, indeed, the methods that we use are diverse: they include, for example, exterior algebra and probabilistic arguments.

Let us indicate how this work is organized. It is split into separate parts, each being a self-contained unit dealing with a particular feature. We tried as far as possible to ensure that each part can be read independently of the others. Please note that each part comes with its own introduction which can be consulted for further information.

Part I: Saturated Hypergraphs

Here we consider the notion of saturation. Let F be a family of forbidden kgraphs, that is, k-uniform set systems. A maximal k-graph G not containing any F ∈ F as a subgraph is called F-saturated. We will be interested in sat(n, F), the minimal number of edges that an F-saturated graph of order n can have. These types of questions were considered as early as the late 40s by Zykov [Zyk49], and by many other mathematicians henceforth.

However, there has been no good general upper bound on the sat-function. Tuza [Tuz86] (also an unpublished conjecture of Bollobás) conjectured that

sat(n, F) = O(n^{k−1}), for any fixed k-graph F.  (1)

While the conjecture was proved for k = 2 by Kaszonyi and Tuza [KT86], and all particular examples confirmed its validity, it was not even known whether generally sat(n, F) = o(n^{k}) for k ≥ 3. In Section 3 we verify this conjecture by showing that sat(n, F) = O(n^{k−1}) for all finite and certain infinite families F of k-graphs.

Different variations of the principle are presented in Section 4: we define the notion of saturation for different graph-like structures and investigate whether a form of (1) holds. While the technique of Section 3 extends to directed cycle-free graphs, ordered graphs, and layered graphs, we had to invent a new method in order to prove (1) for the class of k-row rectangular matrices.

In Subsection 4.4 we consider problems of the following type. Given a forbidden family, we say that a graph G kills an edge E ∈ E(\overline{G}) if the addition of E to G creates a forbidden subgraph. What is the maximal number of killed edges if G has a given order and size? We settle these problems for complete 2-graphs, which extends a theorem of Erdős, Hajnal and Moon [EHM64] who computed sat(n, K_{m}^{2}).

The sat-function is hard to handle: it lacks many natural regularity properties. For example, Kaszonyi and Tuza [KT86] showed that it is not monotone. In Section 5 we amplify their example: we construct, for any constant d, a 2graph F = F (d) such that sat(n, F) < sat(n ± 1, F) − d for a periodic series of values of n. Furthermore, we demonstrate a finite family F of 2-graphs for which the ratio sat(n, F)/n does not tend to a limit, which is rather unexpected and counterintuitive.

Specific instances of forbidden graphs are considered in Section 6.

We asymptotically compute sat(n, S_{m}^{k}), thus extending a result of Erdős, Füredi and Tuza [EFT91] who did the task for S_{k+1}^{k}. (The generalized star S_{m}^{k} is the k-graph on m vertices consisting of all k-tuples containing a given vertex.)

The triangular family T_{k} consists of all k-graphs of size 3 in which the symmetric difference of some two edges is contained in the third one. We prove that sat(n, T_{k}) = n − O(log n), k ≥ 3, and sat(n, T_{3}) = n − 2.

We show that, for any K_{m}-saturated graph G, the number of edges spanned by the set {x ∈ V (G) : d(x) ≤ a} is at most a^{2(m−2)a+o(ma)}, a function not depending on n = v(G). We deduce that G has at least ln + O(\frac{n log log n}{log n}) edges, if the minimal degree of G is l ≥ m − 1. Another consequence is a sharper form of one result by Alon, Erdős, Holzman and Krivelevich [AEHK96, Theorem 2].

The following problem is, in fact, an instance of a sat-type problem. Suppose that we try to construct designs by adding, one by one and as long as possible, k-edges so that each t-set is covered by at most λ edges. What is the worst case, that is, how small the eventual system can be? We solve asymptotically this problem for t = 2 and establish some connections with Turán numbers for general t.

Part II: Weakly Saturated Hypergraphs

A notion related to that of saturation is weak saturation which we consider in Part II. A k-graph G is weakly F-saturated if we can add one by one all missing edges to G so that every time at least one new forbidden subgraph appears; we are interested in w-sat(n, F), the minimal size of a such graph G on n vertices.

These questions were first considered by Bollobás [Bol67c] who made a conjecture on complete graphs. The conjecture was verified by a number of people who computed w-sat(n, K_{m}^{k}): Frankl [Fra82], Kalai [Kal84, Kal85]; the result is implicit in Lovász [Lov77]; cf. also Alon [Alo85]. They all applied some form of dependence in order to derive the formula. This approach was most clearly formulated by Kalai [Kal85]: if we have a matroid M on [n]^{(k)} such that any F ∈ F is a circuit, then w-sat(n, F) ≥ R_{M}([n]^{(k)}), the rank of M.

Usually, it is easy to construct a right example of minimum G ∈ w-SAT(n, F) for a given F, but it is hard to prove that this G is indeed extremal. So, the above approach is helpful but it is not clear at all how to search for a suitable matroid M.

Here we suggest two deterministic candidates for M to consider, provided we have an example of G_{n} ∈ w-SAT(n, F). For this purpose we utilize gross and count matroids which are defined in Section 8. The construction of a gross matroid was exploited by Kalai [Kal90], but for other purposes. Our count matroids form a new family of matroids, considerably and naturally extending the count matroid of White and Whiteley [WW84].

If one of our approaches works, then G is indeed extremal and we say that we have a G-proof or a C-proof respectively. Thus, we have two sufficient criteria for G ∈ w-SAT(n, F) to be minimal. Unfortunately, these criteria are not generally necessary, but using them (and the related g/g′-proof technique) we can prove the following results.

Given sequences of integers s = (s_{1}, …, s_{t}) and k = (k_{1}, …, k_{t}), the pyramid P (s; k) is the k-graph, k = k_{1} + … + k_{t}, with vertex set being the disjoint union S_{1} ∪…∪S_{t}, |S_{i}| = s_{i}, and with the edge set consisting of those k-subsets which, for every i ∈ [t], intersect S_{1} ∪ … ∪ S_{i} in at least k_{1} + … + k_{i} vertices. This is a rather general definition: as partial cases we obtain complete graphs and generalized stars.

In Subsection 10.1 we compute w-sat(n, P (s; k)) for all feasible values of parameters. A partial case of this result proves the conjecture by Tuza [Tuz88, Conjecture 7] that w-sat(n, H_{k}(k + 1, l)) = \binom{n−k+l−2}{l−2}, n ≥ k + 1 ≥ l ≥ 2. (The uniform family H_{k}(m, l) consists of all k-graphs with m vertices and l edges.)

In Subsection 10.2 we present some further results about weakly H_{k}(m, l)- saturated graphs: we make a general conjecture and verify it for a number of parameters. In certain cases we characterize all extremal graphs, in particular answering a question by Erdős, Füredi and Tuza [EFT91] (who verified Tuza’s conjecture for l = 3).

The cone cn(F) of a k-graph F is obtained by adding an extra vertex x plus all \binom{v(F)}{k−1} edges containing x. Our more general results of Section 11 imply that cones ‘preserve’ G/g/g′-proofs under certain covering conditions. This means that if we know the w-sat-function for certain graphs by applying a G/g/g′- proof, then we know it for the graphs obtained by the application of the cone operator. For example, for 2-graphs we can compute w-sat(n, K_{l} + F), where for F we can take a star, an odd cycle, a path, a matching, and many other graphs.

In Section 12 we define join, another operation on graphs, and prove among other things that joins always preserve G/g-proofs. As a special case, we deduce the result of Alon [Alo85] who computed the w-sat-function for joins of complete hypergraphs.

Part III: Chain Decompositions

A chain decomposition of a poset P is a partition of P into disjoint chains (that is, linearly ordered subsets). Minimum chain decompositions have many applications and are extensively studied.

In this part we consider the minimal size of an edge decomposition which is a collection of skipless chains such that any pair x ⋖ y (x is covered by y) belongs to exactly one chain. (A chain C is skipless if no element of P \C can be inserted between some two elements of C.) It is easy to see that edge decompositions of P correspond to skipless chain decompositions of the line poset L(P) whose vertex set is {(x, y) : x, y ∈ P, x ⋖ y}, and (x, y) < (x′, y′) in L(P) if y ≤ x′ in P.

In Section 14 we present a few min-max theorems. Our more general theorem implies that the minimal size of a skipless chain decomposition of P equals the maximal value of |A| − |B| taken over all pairs of disjoint sets A, B ⊂ P such that any skipless chain containing two elements from A intersects B. Surprisingly enough, this fundamental theorem turned out to be a new result. Our proof utilizes the linear programming method of Dantzig and Hoffman [DH56]. It was considerably simplified by Graham Brightwell who replaced the linear programming argument by an easy application of Hall’s theorem. We present both these proofs.

The minimal size of an edge decomposition of P can be deduced as a corollary, but we provide a short and direct proof.

Hence, our basic question is generally completely answered, but we can ask whether there is an edge decomposition with some extra properties. Of course, one can consider these problems for many different posets and impose many different restrictions. But as our theme is extremal set systems, we investigate B_{n}, the poset of subsets of an n-set ordered by inclusion, and ask whether we can require that all chains are symmetric. (A skipless chain A_{1} ⊂ … ⊂ A_{k} of B_{n} is symmetric if |A_{i}| + |A_{k−i+1}| = n, 1 ≤ i ≤ k.) Note that any symmetric edge decomposition of B_{n} has the minimal size.

In fact, the general results of Anderson [And67] and Griggs [Gri77] imply the existence of a symmetric edge decomposition of B_{n}. However, their proofs are non-constructive, so in Section 15 we provide an explicit construction.

Our decomposition has some extra properties and interesting applications, see Section 16. In brief, we give estimates of the number of antichains in L(B_{n}), construct a pair of orthogonal skipless chain decompositions of L(B_{n}), present some applications to storing and searching records in a database, and solve one numerical problem.

In Section 17 we characterize line posets in terms of forbidden configurations and point out which information determines and can be reconstructed from the line poset. (This resembles Beineke’s [Bei68] characterization of line graphs.)

Part IV: Enumeration Results for Trees

Here we consider and enumerate different tree-like structures. Strictly speaking, such problems belong to enumerative, rather than to extremal, graph theory, but we include these results because we believe that the proofs are short and nice.

The notion of a tree and its different extensions to k-graphs, that is, kuniform set systems, play an important role in discrete mathematics and computer science. We will dwell upon the following, rather general, definition suggested independently by Dewdney [Dew74] and Beineke and Pippert [BP77].

A k-graph is called a (k, m)-tree if it can be obtained from a single edge by consecutively adding edges so that every new edge contains k − m new vertices while its remaining m vertices are covered by an already existing edge.

The problem of counting (m + 1, m)-trees which are known in the literature as m-trees, received great attention and was completely settled by Beineke and Pippert [BP69] and Moon [Moo69]. Later, different bijective proofs for m-trees appeared as well, see [RR70, Foa71, GI75, ES88, Che93].

Here we enumerate vertex labelled (k, m)-trees. We present two different proofs. The proof of Section 19 is inductive, that is, we write a recurrence relation for the number of trees and prove our formula by induction.

In Subsection 20.2 we exhibit an explicit bijection between the set of rooted vertex labelled trees of given size and a trivially simple set; it is based on the ideas of Foata [Foa71]. This method can be applied to enumerate other tree-like structures. For example, we enumerate vertex labelled k-gon trees. A k-gon tree is obtain from a k-gon (that is, a k-cycle) by consecutively adding k-gons along an existing edge, see e.g. [CL85, Whi88, Pen93, KT96]. In order not to repeat the same portions of proof twice, we present a more general result which includes both (k, m)-trees and k-gon trees as partial cases.

In Subsection 20.3 we present a bijection for edge labelled (2, 1)-trees, answering a question posed by Cameron [Cam95]. Unfortunately, we do not know any direct bijection enumerating edge labelled (k, m)-trees for general k, m.

Part V: Large Degrees in Subgraphs

Erdős [Erd81], see also [Chu97, Erd99], conjectured that for n ≥ 3 any graph with fewer than \binom{2n+1}{2} − \binom{n}{2} = \frac{3n(n+1)}{2} edges is a union of a bipartite graph and a graph with maximum degree less than n. All research carried in this part is motivated by this conjecture which is disproved here.

The conjectured value arises from the consideration of P_{n+1,n} = K_{n+1} + E_{n} which does not admit the above representation. In fact, this graph has a stronger property, namely P_{n+1,n} → (K_{1,n}, K_{3}): for any blue-red colouring of the edge set of P_{n+1,n} we necessarily have either a blue star K_{1,n} or a red triangle. Thus, if Erdős’ conjecture were true, it would give the same value for the size Ramsey number r̂(K_{1,n}, K_{3}) = min{e(G) : G → (K_{1,n}, K_{3})}. Apparently, the computation of r̂(K_{1,n}, K_{3}) was the original motivation for the conjecture.

In Section 22 we show, however, that

r̂(K_{1,n}, K_{3}) < n^{2} + \sqrt{2} n^{3/2} + n, n ≥ 1,

by demonstrating an explicit construction. This disproves Erdős’ conjecture which, in fact, fails for all n ≥ 5. On the other hand, we prove that any graph with n^{2} + (0.577 + o(1)) n^{3/2} edges is a union of a bipartite graph and a graph with maximum degree less than n, which of course implies that this number is a lower bound for r̂(K_{1,n}, K_{3}).

There were different attempts to prove the conjecture, by different mathematicians, which resulted in new interesting directions of research.

For example, as reported in [Erd99], Erdős and Faudree [EF99] consider the minimal size of a graph G such that if G is a union of two graphs, one having maximal degree less than n, then the other contains all odd cycles C_{m} with 3 ≤ m ≤ n − 3. In Subsection 22.3 we demonstrate a graph G of size (1 + ε)n^{2}, for any given constant ε > 0, such that, for any blue-red colouring of G without a blue K_{1,n}, we have red cycles of all lengths (odd and even) between 3 and cn, where c = c(ε) > 0 does not depend on n.

The following problem, which was introduced by Erdős, Reid, Schelp and Staton [ERSS96], is also motivated by Erdős’ conjecture.

For positive integers n, k, j with k ≥ j, let M(n, k, j) consist of all graphs G of order n + k such that every (n + j)-subset of V (G) spans a graph with maximum degree at least n. The question is to compute

m(n, k, j) = min{e(G) : G ∈ M(n, k, j)}.

Erdős et al [ERSS96, Conjecture 1] conjectured that, for n ≥ k ≥ j ≥ 1 and n ≥ 3, we have

m(n, k, j) = (k − j + 1)n + \binom{k − j + 1}{2}.  (2)

This value arises from the consideration of P_{k−j+1,n}⊔E_{j−1}. Erdős et al [ERSS96, Theorem 3] proved that (2) is true if j = 1 or if j ≥ 2 and

n ≥ max (j(k − j), \binom{k−j+2}{2}) .  (3)

In Section 23 we demonstrate a constructive counterexample to (2) for n ≤ (j − 2)(k − j). On the other hand, we show that (2) is true if

n ≥ max ((j + \frac{1}{2}) (k − j) + \frac{j+k}{4j−2}, 14) ,

which improves (3) for j ⪅ k/3. This shows that j(k−j) is roughly the threshold on n when the obvious construction leading to (2) fails to be extremal. Some other constructions are presented.

In Section 24 we consider the following related problem. Let B(n, m) consist of all graphs such that for any partition V (G) = A ∪ B either Δ(G[A]) ≥ n or Δ(G[B]) ≥ m (or both). We are interested in the bisplit function

b(n, m) = min{e(G) : G ∈ B(n, m)}.

Clearly, b(n, n) is precisely the function investigated in Erdős’ conjecture, which was the original motivation for introducing the ‘off-diagonal’ numbers b(n, m).

We compute this function asymptotically when m = min(n, m) is large:

b(n, m) = 2nm − m^{2} + o(m)n.

In the extreme case, when m ≥ 1 is fixed, we can prove only that the numbers b(n, m), n ∈ ℕ, lie between two functions linear in n with slopes 2m + 1 and 2m + \sqrt{2m} + \frac{5}{2}.

We prove that b(n, 1) = 4n − 2 for n ≥ 8 (and characterize all extremal graphs) and that b(n, 2) = 6n + O(1). As the reader will see, the proofs are rather lengthy and require consideration of many cases. This indicates that the computation of lim_{n→∞} b(n, m)/n for any fixed m (if the limit exists) is perhaps a hard task.

Notation

Let us indicate some notation that we use. The relation A ⊂ B does not exclude A = B; the strict inclusion is denoted as A ⊊ B. Any unfamiliar term (e.g. pyramid) should be identifiable via the index.

[m, n] = {m, m + 1, …, n}; [n] = {1, 2, …, n}
A^{(r)} = {B ⊂ A : |B| = r}
ℝ/ℚ/ℤ/ℕ  the sets of reals/rationals/integers/positive integers
f = Θ(g) ⇔ ∃c_{1}, c_{2} > 0 ∃n_{0} ∀n ≥ n_{0} c_{1}g(n) ≤ |f(n)| ≤ c_{2}g(n)
f = O(g) ⇔ ∃c > 0 ∃n_{0} ∀n ≥ n_{0} |f(n)| ≤ cg(n)
f = o(g) ⇔ ∀c > 0 ∃n_{0} ∀n ≥ n_{0} |f(n)| ≤ cg(n)
s_{A} = ∑_{i∈A} s_{i}, given reals s_{1}, …, s_{n} and A ⊂ [n]
B_{A} = ∪_{i∈A}B_{i}, given sets B_{1}, …, B_{n} and A ⊂ [n]
V (G)  the vertex set of G
v(G) = |V (G)|  the order of G
E(G)  the edge set of G
e(G) = |E(G)|  the size of G
\overline{G}  the complement of G
G[A]  the subgraph induced by A ⊂ V (G)
α(G)  the independence number of G
d(x) = |{E ∈ E(G) : E ∋ x}|, x ∈ V (G)
Δ(G)/δ(G)  the maximal/minimal degree of G
Γ_{A}(x) = {y ∈ A : {x, y} ∈ E(G)} , 2-graph G; Γ(x) = Γ_{V (G)}(x)
d_{A}(x) = |Γ_{A}(x)|, x ∈ V (G), A ⊂ V (G), 2-graph G
mG  m disjoint copies of G
C_{m}  the m-cycle
E_{m}  the empty graph of order m
K^{k}(A)  the complete k-graph on a set A
K_{m}^{k}  the complete k-graph of order m; K_{m} = K_{m}^{2}
K_{m,n}  the complete bipartite graph
P_{m,n} = K_{m} + E_{n}
P (s; k)  the pyramid
S_{m}^{k}  the k-star of order m

Contents

I Saturated Hypergraphs 1

1 Introduction 1 1.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Survey 3 2.1 General Families . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Construction 7

4 Variations 10

4.1 Graphs with Oriented Edges . . . . . . . . . . . . . . . . . . . . . 11 4.1.1 Directed Hypergraphs . . . . . . . . . . . . . . . . . . . . 11 4.1.2 Ordered Hypergraphs . . . . . . . . . . . . . . . . . . . . 14

4.2 Layered Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Forbidden Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.4 Edge Killers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Irregularities 26

6 Specific Classes 30 6.1 Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.2 Sub-Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3 Triangular Families . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.4 K_{m}-Saturated Graphs . . . . . . . . . . . . . . . . . . . . . . . . 43

II Weakly Saturated Hypergraphs 47

7 Introduction 47 7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.3 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8 Matroids 52

8.1 Gross Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8.1.1 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . 52 8.1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.2 Count Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.2.2 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

9 Proof Techniques 60

10 Specific Classes 63 10.1 Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.2 Uniform Families . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

10.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . 70 10.2.2 Applications of Count Matroids . . . . . . . . . . . . . . . 71 10.2.3 Applications of Gross Matroids . . . . . . . . . . . . . . . 76

10.3 Miscellaneous Graphs . . . . . . . . . . . . . . . . . . . . . . . . 77

11 Cones 84

12 Joins 89

III Chain Decompositions 94

13 Introduction 94 13.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 13.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

14 Skipless Chain Decompositions 96

15 Symmetric Edge Partitions of Cubes 101

16 Applications of the Partition D_{n} 105 16.1 On the Number of Antichains in L(B_{n}) . . . . . . . . . . . . . . . 105 16.2 Orthogonal Partitions of L(B_{n}) . . . . . . . . . . . . . . . . . . . 105 16.3 A Storage and Retrieval Problem . . . . . . . . . . . . . . . . . . 106 16.4 One Numerical Problem . . . . . . . . . . . . . . . . . . . . . . . 108

17 Characterization of Line Posets 110

IV Enumeration Results for Trees 113

18 Introduction 113

19 Inductive Proof 114

20 Bijective Proofs 115 20.1 Foata’s Bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 20.2 H-Built-Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 20.3 Edge Labelled Trees . . . . . . . . . . . . . . . . . . . . . . . . . 119

V Large Degrees in Subgraphs 121

21 Introduction 121

22 Triangle-vs-Star Size Ramsey Number 123 22.1 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 22.2 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 22.3 Cycles of Consecutive Lengths . . . . . . . . . . . . . . . . . . . . 129

23 Removing Vertices 131 23.1 Some Constructions . . . . . . . . . . . . . . . . . . . . . . . . . 131 23.2 Improving Condition (107) . . . . . . . . . . . . . . . . . . . . . . 133

24 Splitting into Parts 134 24.1 General Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 24.2 Small Fixed m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Part I

Saturated Hypergraphs

1 Introduction

1.1 Discussion

Many combinatorial structures (especially graphs) have proved to be very useful in other branches of human knowledge where discrete models play more and more important role with the advance of computers. A fairly typical problem is, given a class C of allowed graphs (for example, those whose structure is compatible with the requirements of the used discrete model), to minimize/maximize a certain parameter.

In many natural cases, C can be described by naming a family F of forbidden subgraphs so that a graph belongs to C if and only if it is F-free, that is, if it does not contain any F ∈ F as a subgraph. In this case, C can be also specified by listing the family SAT(F) of all F-saturated graphs, that is, maximal F-free graphs; clearly,

C = {H : H ⊂ G for some G ∈ SAT(F)}

and, instead of considering the whole of C, we can restrict ourselves only to SAT(F), especially that many extremal parameters of C can be more quickly determined from SAT(F).

Two related families are m-SAT(F) and w-SAT(F): G ∈ m-SAT(n, F) if the addition any new edge to G creates at least one new forbidden subgraph (then we call G monotonically F-saturated); G ∈ w-SAT(n, F) if we can add all missing edges, one by one in some order, so that every edge creates a new forbidden subgraph (then we call G weakly F-saturated). Note that we do not require here that G is F-free.

For example, the Turán-type problem studies the maximal size of an Ffree graph of a given order. This is clearly equal to the maximal size of an F-saturated graph of a given order.

In Parts I and II we consider the sat-type problems which ask about the minimal size of a (weakly/monotonically) F-saturated graph of a given order.

The Turán-problem and the sat-problem happen to be rather different in nature. The former is perhaps more important in applications although many real life situations lead to sat-type questions.

For example, monotonically K_{3}-saturated graphs are precisely diameter-2 graphs. This problem has the following interpretation: there are n airports; we can connect some pairs by a flight and we want to assure the possibility to fly from each airport to any other one by changing the plane at most once. Clearly, the minimal number of connections is n−1 and this is achieved if and only if one airport is connected to every other. (This may be not a perfect solution and we may impose some extra conditions: e.g. some restriction on the maximal degree, etc.) If we weaken the requirement by allowing any number of flight changes, then we obtain weakly K_{3}-saturated (i.e. connected) graphs and the minimal size is again n − 1 but we have many extremal graphs.

In this part (and Part II) we try to present a unified treatment of these, sat-type, questions. The above definitions are applied to k-graphs (k-uniform set systems) which are the main object of our consideration. Also, we present different variations of the principle and make a few excursions into some related areas (for example, the forbidden submatrix problem). Section 2 briefly surveys known results on the topic including those proved here. But before we proceed, let us give all necessary definitions.

1.2 Definitions

Let F be a family k-graphs (that is, k-uniform set systems) which are usually referred to as forbidden. A k-graph G is called F-admissible (or F-free) if it does not contain any F ∈ F as a subgraph.

We say that G is F-saturated, denoted G ∈ SAT(n, F), if it is a maximal F-free k-graph with n vertices. We are mainly interested in

sat(n, F) = min{e(G) : G ∈ SAT(n, F)},  (4)

the minimal number of edges in an F-saturated graph of order n.

The following auxiliary notion is helpful: G is called monotonically (or strongly) F-saturated, denoted G ∈ m-SAT(n, F), n = v(G), if the addition of any new edge to G creates at least one extra F -subgraph, some F ∈ F. Note that we do not require that G is F-admissible.

Clearly, SAT(n, F) = {G ∈ m-SAT(n, F) : G is F-free}, so sat(n, F) ≥ m-sat(n, F), where

m-sat(n, F) = min{e(G) : G ∈ m-SAT(n, F)}.

For a graph F, we denote SAT(n, F) = SAT(n, {F }), etc.

2 Survey

Here is a brief but comprehensive (to the best of the author’s knowledge) survey of known results related to (strong) saturation. Also, we indicate all interesting results proved in this part.

2.1 General Families

Not much is known about sat(n, F) for a general F. Kászonyi and Tuza [KT86] showed that, for any family F of 2-graphs, including all infinite families, we have sat(n, F) = O(n). Tuza [Tuz92] showed that, for any fixed k-graph F,

m-sat(n, F) = Θ(n^{d(F)}).  (5)

Here d(F) ∈ [0, k − 1] is what Tuza calls the local density of F:

d(F) = min{d(E) : E ∈ E(F)},  (6)

where the density d(E) of an F -edge E is max {|E ∩ E′| : E′ ∈ E(F) E′ ≠ E}.

Clearly, in terms of constructive upper bounds, SAT is more restrictive than m-SAT. Thus, it is not surprising that, up to now, there were no good upper bounds on sat(n, F) for a general k-graph F. Tuza [Tuz86, Tuz88] (also an unpublished conjecture of Bollobás) conjectured that, for any fixed k-graph F, sat(n, F) = O(n^{k−1}).

In Section 3 we show that

sat(n, F) = O(n^{k−1})  (7)

for all finite and certain infinite families F, which, of course, proves this conjecture. Our proof is constructive.

In Section 4 we try to extend the notion of saturation to different structures connected to hypergraphs and every time we ask whether the analogue of estimate (7) is valid. Although the estimate is not true for simple directed graphs, we show that (7) is valid for all finite families of cycle-free directed k-graphs and for ordered k-graphs. Furthermore, the estimate sat(n, F) = O(n) is true for any family F of cycle-free or ordered 2-graphs.

In Subsection 4.2 we consider similar question for structures that we call layered graphs and show that a form of (7) holds here. Also, we show that, for the class of layered (1, 1)-graphs (that is, bipartite graphs), the size of any minimum F-saturated graph is bounded by a linear function of its order for any forbidden family F.

In Subsection 4.3 we consider the sat-type problems for the class of rectangular matrices, for which the dual (Turán-type) problems are well studied. We show that for any family F of forbidden k-row matrices sat(n, F) = O(n^{k−1}).

Although the notion of saturation was considered as early as the late 40s by Zykov [Zyk49], the theory does not seem to be well developed. This might be the case because minimum saturated graphs are hard to handle. For example, as demonstrated by Kászonyi and Tuza [KT86], the sat-function lacks many natural regularity properties; in Section 5 we provide further examples.

Answering a question by Tuza [Tuz92] we exhibit an example of connected 2-graphs H ⊂ F of the same order such that sat(n, H) > sat(n, F) for all large n. (Of course, it is ‘natural’ to expect the converse inequality.)

Among other things, we demonstrate, for any fixed d > 0, a 2-graph F = F (d) such that

sat(n, F) < sat(n ± 1, F) − d,

for a periodic series of values of n.

Tuza [Tuz88] conjectured that, for any 2-graph F, the limit lim sat(n, F)/n exists. Of course, a number of similar questions arise for k-graphs as well. Unfortunately, there is not much progress in this direction.

Truszczynski and Tuza [TT91], characterized those 2-graphs F for which c = lim sat(n, F)/n exists and is smaller than 1; then, in fact, c = 1 − 1/p, p ∈ ℕ.

In Section 5 we demonstrate a finite family F of 2-graphs for which the limit lim sat(n, F)/n does not exist.

In the literature, there are many different variations on the topic; one possibility is to consider minimum saturated graphs (most frequently K_{m}^{2}-saturated) with some extra restrictions, for example, on degrees (Hajnal [Haj65], Hanson and Seyffarth [HS84], Duffus and Hanson [DH86], Erdős and Holzman [EH94], Füredi and Seress [FS94], Alon et al [AEHK96]), chromatic number (Hanson and Toft [HT91]), etc. Hanson and Toft [HT87] consider edge-coloured saturated graphs.

2.2 Particular Cases

Erdős, Hajnal and Moon [EHM64] via an inductive argument and contractions computed the sat-function for all complete 2-graphs. Bollobás [Bol65] introduced the powerful weight method and proved that

sat(n, K_{m}^{k}) = \binom{n}{k} − \binom{n − m + k}{k}, n ≥ m > k.  (8)

The cases of equality were characterized in both papers.

We show that, for any K_{m}-saturated graph G, the number of edges spanned by the set {x ∈ V (G) : d(x) ≤ a} is bounded by a^{2(m−2)a+o(ma)}, a function of a and m only. We deduce that G has at least ln + O(\frac{n log log n}{log n}) edges, n = v(G), if the minimal degree of G is l ≥ m − 1. Another consequence is a sharper form of one result by Alon, Erdős, Holzman and Krivelevich [AEHK96, Theorem 2]. Please refer to Subsection 6.4 for details.

The star S_{m}^{k} has m vertices and consists of k-tuples containing a fixed vertex. The uniform family H_{k}(m, l) consists of all k-graphs of order m and size l. Erdős, Füredi and Tuza [EFT91] determined the exact sat-values for H_{3}(6, 3) and H_{3}(4, 3) = S_{4}^{3} and described the cases of equality. Also, they found asymptotic values for H_{k}(k + 1, k) = S_{k+1}^{k}. In Subsection 6.1 we extend the last result by computing asymptotically sat(n, S_{m}^{r}) for all possible r and m.

In Subsection 6.2 we define a t-(v, k, λ)-sub-design G as a maximal k-graph of order n such that no t-set is covered by more than λ edges. (Sub-designs naturally arise when we try to construct designs by consecutively adding edges as long as possible.) If we let D = D(λ, k, t) be the family of all k-graphs with λ + 1 edges sharing at least t common vertices then SAT(n, D) is the family of all sub-designs of order n. We compute exactly sat(n, D(λ, k, t)) for t = 1 and any λ, k, n (except for a few small values of n) and (asymptotically) for t = 2 and any fixed λ, k. In the general case t ≥ 3 we deduce some lower bounds and establish connections with the Turán problem for complete hypergraphs.

In Subsection 6.3 we forbid 3 edges such that the symmetric difference of some two edges is contained in the third one and compute asymptotically the corresponding sat-function. (For 3-graphs, we find the exact value.)

Erdős and Gallai [EG61] showed that mK_{3}^{2} is the (unique) minimum graph in SAT(n, mK_{2}^{2}) for n ≥ 3m. (By mF we denote the union m disjoint copies of F.) The case of mK_{k}^{k}, k ≥ 3, is harder. Many authors present different lower and upper bounds on sat(n, 2K_{k}^{k}) for specific k. The best known general bounds seem to be sat(n, 2K_{k}^{k}) ≤ k^{5}, k ≥ 1, by Blokhuis [Blo87], and sat(n, 2K_{k}^{k}) ≥ 3k, k ≥ 4, by Dow et al [DDFL85].

Wessel [Wes66, Wes67] and Bollobás [Bol67b, Bol67a] computed independently the sat-function and characterized extremal graphs for all complete bipartite graphs in the class of bipartite, that is, (1, 1)-layered, graphs.

Concerning 2-graphs, Kászonyi and Tuza [KT86] found the complete answer for all paths and stars. The situation for cycles looks rather complicated. Of course, the case C_{3} = K_{3}^{2} is known. Ollman [Oll72] proved that sat(n, C_{4}) = ⌊(3n−5)/2⌋ and all extremal graphs were described by Tuza [Tuz89]. According to a recent paper by Barefoot et al [BCE^{+}96], for every k ≥ 5, we know the exact values of sat(n, C_{k}) only for finitely many values of n although some general bounds are available.

A result of Bondy [Bon72b] implies that

sat(n, C_{n}) ≥ ⌈3n/2⌉.  (9)

There was a great amount of work invested in computing this function exactly (Isaacs [Isa75], Clark et al [CE83, CCES86, CES92]) until the computation was completely finished by Xiaohui et al [XWCY97] (with final touches made by computer search). In fact, estimate (9) is sharp for all even n ≥ 20 and all odd n ≥ 17.

Füredi et al [FHPZ98] considered digraphs and showed that sat(n, \overrightarrow{C_{3}}) = (1 + o(1))n log_{2} n. (Here \overrightarrow{C_{3}} denotes the directed 3-cycle.)

In Subsection 4.4 we investigate the maximal number of edges which cannot be F-freely added to G, given v(G) and e(G). We settle this problem (with a description of all extremal graphs) for complete 2-graphs, which extends the already mentioned result of Erdős, Hajnal and Moon [EHM64] who computed sat(n, K_{m}^{2}).

3 Construction

Here we demonstrate some constructive upper bounds on sat(n, F) for a general family F which, in particular, imply the conjecture of Tuza [Tuz86] (also conjectured by Bollobás, unpublished) that, for any k-graph F,

sat(n, F) = O(n^{k−1}).  (10)

Note that we cannot replace k − 1 by a smaller exponent in (10) if we want the estimate to be valid for every k-graph F; this follows, for example, from formula (8).

Kászonyi and Tuza [KT86] proved that sat(n, F) = O(n), for any family F of forbidden 2-graphs, including infinite families; this verifies (10) for k =

2. However, there has been no progress in proving (10) for k ≥ 3 and the conjecture is mentioned in a few different papers, e.g. in [Tuz88, EFT91, Tuz92, Fra95]. Also, the importance of estimate (10) might be indicated by the fact that Bollobás [Bol95], in his authoritative survey of the whole of extremal graph theory, gives two different proofs of sat(n, F) = O(n) for 2-graphs.

Let us present some general construction of H ∈ sat(n, F) which implies (10); this result appears in [Pik99d].

For a k-graph H, we say that A ⊂ V (H) is independent if it does not span an edge in H, that is, A^{(k)} ∩ E(H) = ∅.

Theorem 1 Let F be a family of k-graphs. Suppose that there is s ∈ ℕ such that no F ∈ F contains an independent set A ⊂ V (F) of order s + 1 which can be covered by a union of F -edges sharing a common vertex outside A. Then, for any n,

sat(n, F) < (s′ − s + 2^{k−1}(s − 1)) \binom{n}{k − 1},  (11)

where s′ = min{v(F) : F ∈ F}.

Proof. It is enough to construct a graph H ∈ SAT(n, F) whose size does not exceed the stated bound. Our construction will be by means of an algorithm.

Our algorithm works in the following way. Let us agree that the vertex set is X = [n] with the usual ordering. Given x ∈ X and B ⊂ X, we write B < x if every vertex in B is smaller than x. By U_{x} = {y ∈ X : y > x} we denote the upper shadow of x and in the obvious way we define the lower shadow L_{x}. If |B| ≤ k, say B consists of elements b_{1} < … < b_{i}, i ≤ k, then we define its tail

T_{B} = {{b_{1}, …, b_{i}, x_{i+1}, …, x_{k}} : b_{i} < x_{i+1} < … < x_{k}} ⊂ X^{(k)}. (12)

We construct an F-saturated graph H by starting with the empty hypergraph H on X and adding to H one by one certain families of edges until we obtain H ∈ SAT(n, F).

The algorithm is rather simple. We take, one by one in order, the vertices of X. For every vertex x, we consider all of the i-subsets of L_{x}, beginning with i = 0 and increasing i until i = k − 1. For every such subset A < x, we consider T_{B}, B = A ∪ {x}, which is, by the definition, the family of k-subsets having B as an initial segment. If at this moment T_{B} ⊄ E(H) and the addition of T_{B} to the edge set of H does not create any forbidden subgraph, we add T_{B} to H. This is a crucial feature of the algorithm: for every x and A we either add all of T_{B} or we add nothing.

Another important detail is the order of the steps. The outermost cycle has x increasing from 1 to n. The next cycle runs for i increasing from 0 to k − 1. In the innermost cycle we consider all i-subsets of L_{x} and here we are free to choose them in any order, but for uniformity let us agree that we use here the colex order.

In the course of the algorithm we define, on the vertex set X, auxiliary hypergraphs H_{1}, …, H_{n} and G_{1}, …, G_{k} which we need for an estimation of e(H) = |E(H)|. The k-hypergraph H_{x} contains precisely those edges which were added whilst considering vertices from 1 to x inclusive. The i-hypergraph G_{i} contains as edges those i-subsets B for which the set T_{B} was added to H.

We claim that the resulting graph H = H_{n} is an F-saturated graph. Indeed, H is F-admissible, as we were adding edges only if they did not produce any forbidden subgraphs. On the other hand, take any k-subset E not in E(H). We did not use the opportunity to add E to E(H) when x = max E, i = k − 1 and A = E \ {x} (when T_{B} = {E}). The only reason for our not doing so is that the addition of E would have created a forbidden subgraph F. Then certainly, H + E contains F, which shows H ∈ SAT(n, F).

We claim that e(G_{1}) ≤ s′ − 1 and

e(G_{i}) ≤ (s − 1)\binom{n}{i − 1}, i = 2, …, k.  (13)

Assume that for some i ∈ [2, k] the estimate (13) is not true. Then there is some (i − 1)-set V = {v_{1}, …, v_{i−1}}, v_{1} < … < v_{i−1}, which is the initial segment of at least s edges of G_{i}. Let E_{1}, …, E_{s} ∈ E(G_{i}) be s distinct edges containing V as an initial segment, say E_{j} = V ∪ {z_{j}}, j ∈ [s], V < z_{1} < … < z_{s}.

Since E_{1} ∈ E(G_{i}), all edges whose initial segment is E_{1} were added to H at the moment when x = z_{1} and A = V. It follows that V ∉ E(G_{i−1}) for otherwise these edges would have already been present in H. The only reason that we did not add V to E(G_{i−1}) earlier when x = v_{i−1} and A = {v_{1}, …, v_{i−2}} must have been that the hypergraph H′ = H_{v_{i−1}} + T_{V} contains some forbidden subgraph F. Let

Y = {u ∈ U_{v_{i−1}} : u ∈ E for some E ∈ E(F) ∩ T_{V} }.

As U_{v_{i−1}} is an independent set in H′ and each edge in T_{V} contains v_{i−1} the assumptions of the theorem imply that |Y | ≤ s.

By the way algorithm works, any permutation σ of X affecting only the upper shadow U_{z} of a vertex z ∈ X (that is, σ(y) = y for all y ≤ z) is an automorphism of H_{z} because any T_{B} ⊂ X^{(k)} with z ≥ max B is σ-invariant. Applying this remark to z = v_{i−1} we see that we may assume Y ⊂ Z = {z_{1}, …, z_{s}}.

Let E ∈ E(F) \ E(H) ⊂ T_{V} which exists as F ⊄ H. Clearly, E ∩ U_{v_{i−1}} ⊂ Y and E ∈ T_{E_{j}}, where z_{j} = min E ∩ {z_{1}, …, z_{s}}. Since E_{j} ∈ E(G_{i}) we obtain the contradiction E ∈ E(H), so (13) is proved for any i ∈ [2, k].

The case i = 1 does not fall into general scheme of the proof. But it is rather trivial, for if we have at least s′ edges (one-element subsets) in G_{1}, say {v_{1}}, …, {v_{s′}} ∈ E(G_{1}), then these vertices span a complete k-graph in H, because if E ∈ {v_{1}, …, v_{s′}}^{(k)} then E ∈ T_{{min E}} ⊂ E(H). Therefore H contains every k-graph of order s′ which is certainly a contradiction.

Clearly, every edge of G_{i} corresponds to less than \binom{n−i+1}{k−i} edges of H so by (13) we obtain

e(H) − (s′ − s)\binom{n}{k − 1} < (s − 1) ∑_{i=1}^{k} \binom{n − i + 1}{k − i}\binom{n}{i − 1}
= 2^{k−1}(s − 1)\binom{n}{k − 1},

which establishes the theorem.

Remark. Our construction is not generally best possible. For example, for 2K_{2}^{2}, the sat-function equals 3 while our algorithm gives n − 1.

Corollary 2 For any finite family F of k-graphs, sat(n, F) = O(n^{k−1}).

An interesting question which still remains open is the following.

Problem 3 Is the estimate sat(n, F) = O(n^{k−1}) valid for any infinite family F of k-graphs, k ≥ 3? (True for k = 2, see Kászonyi and Tuza [KT86].)

Tuza [Tuz92] made the following (still open) conjecture which is stronger than (10).

Conjecture 4 (Tuza) For any k-graph F we have sat(n, F) = Θ(n^{d(F)}), where d(F) is defined by (6). Probably, the stronger assertion sat(n, F) = cn^{d(F)} + O(n^{d(F)−1}), for some constant c, is also true.

4 Variations

Here we consider sat-type questions for a variety of structures. Note that the notion of a saturated structure can be defined in quite general settings, cf. Tuza [Tuz86].

Suppose that we have a class C of objects with a binary relation ‘⊂’ which is a partial order and a rank function r : C → ℕ such that G ⊂ H implies r(G) ≤ r(H). Given a family F of elements of C, we say that H ∈ C is Fadmissible if H does not contain an F ∈ F as a subobject. Now, let SAT(n, F) be the family of all maximal F-admissible objects of rank n. An object H is called F-saturated if H ∈ SAT(r(H), F).

In some cases, C will be the class of hypergraphs with some additional structure: for G, H ∈ C, r(H) = v(H) and G ⊂ H holds if G is a subgraph of H in a structure-compatible way. Thus, H is F-saturated if it does not contain any forbidden substructure and this fails to be true for any H′ ∈ C strictly containing H and having the same order.

Usually, we will ask whether the estimate

sat(n, F) = O(n^{k−1})  (14)

is true for a general ‘k-graph’ family F and for the appropriately defined satfunction.

4.1 Graphs with Oriented Edges

Here we shall consider, roughly speaking, k-hypergraphs with the additional structure of directed edges.

4.1.1 Directed Hypergraphs

To obtain a directed hypergraph we take a usual hypergraph and on every one of its edges introduce some orientation, that is, a linear order.

In fact, estimate (14) is not generally true in these settings. For example, improving previous results of Katona and Szemerédi [KS67], Füredi, Horak, Pareek and Zhu [FHPZ98] showed that sat(n, C_{3}) ≈ n log_{2} n, where \overrightarrow{C_{3}} denotes the directed 3-cycle: E(\overrightarrow{C_{3}}) = \{(1, 2), (2, 3), (3, 1)\}.

But the situation is different if we consider cycle-free (or acyclic) hypergraphs, that is, those not containing a cycle which is, by definition, an alternating sequence of vertices and edges

(x_{1}, E_{1}, x_{2}, E_{2}, …, x_{l}, E_{l}, x_{l+1} = x_{1})

such that x_{i} precedes x_{i+1} in E_{i}. Equivalently, a graph H is cycle-free if we can order its vertices in a way compatible with the ordering of its edges.

By definition, H is F-saturated if no F ∈ F is a subgraph of H but the addition of any new (ordered) edge to G creates either a forbidden subgraph or an oriented cycle. We say that A ⊂ V (F) is independent if no edge of F lies within A.

Theorem 5 In the class of the cycle-free k-graphs, let F be a forbidden family such that the size of any independent set A ⊂ F ∈ F covered by a union of F -edges sharing a vertex outside A, is bounded. Then sat(n, F) = O(n^{k−1}).

Proof. We proceed essentially in the same way as in the proof of Theorem 1, but there are new technicalities.

Consider one by one x ∈ X = [n], i = 0, …, k−1, A ∈ L_{x}^{(i)}. Let B = A∪{x} and let T_{B} be defined by (12). An orientation of the edges in T_{B} is called symmetric if any order preserving injections f, g : [k] → [n] with f([k]), g([k]) ∈ T_{B} induce identical orientations of [k].

If T_{B} ⊄ E(H) (as unoriented k-tuples) and there exists a symmetric orientation of T_{B} such that H +T_{B} does not contain a forbidden subgraph or a cycle, then we add T_{B} (with this orientation) to the edge set of H.

That is the algorithm. The obtained hypergraph H does not contain a forbidden configuration. As every k-subset E ⊂ X was tested (for B = E we had T_{B} = {E} and every orientation was symmetric), we conclude that H ∈ SAT(n, F).

As in Theorem 1 we define the auxiliary hypergraphs H_{x} (directed) and G_{i} (undirected). We have to show that e(G_{i}) = O(n^{i−1}).

First, suppose that E(G_{1}) = {{x_{1}}, …, {x_{l}}}, x_{1} < … < x_{l}. One can easily check that, as H is cycle-free, there is no choice for the orientation of the edges of T_{{x_{i}}}, 2 ≤ i ≤ l and H contains the complete cycle-free k-graph on l vertices, which implies l = O(1), as required.

Suppose that e(G_{i}) ≠ O(n^{i−1}), for some 1 < i ≤ k. Then, for some (i − 1)- tuple V ⊂ X, we can find an arbitrarily large set Z = {z_{1}, …, z_{s}} ⊂ U_{x}, x = max V, such that V ∪{z_{i}} ∈ E(G_{i}), i ∈ [s], and the orientation of ∪_{i∈[s]}F_{V ∪{z_{i}}} ⊂ E(H) extends to a symmetric orientation ‘≺’ of T_{V}. As V ∉ E(G_{i−1}) we conclude that H′ = H_{x} + (T_{V} , ≺) contains a forbidden subgraph F or a cycle. If a copy of F is present we follow the proof of Theorem 1. Otherwise let C = (y_{1}, E_{1}, …, y_{l}, E_{l}, y_{l+1} = y_{1}) be a shortest cycle in H′.

We claim that C can be chosen so that |W | ≤ 3k −5, where W = (∪_{i∈[l]}E_{i})∩ U_{x}. Then for s ≥ 3k − 5 we may assume that W ⊂ Z, and the argument of Theorem 1 shows that C ⊂ H, which is a contradiction proving the theorem.

If Y = {y_{1}, …, y_{l}} ⊂ U_{x} then l ≤ 2 and the claim is true. Indeed, there is an i ∈ [l] such that y_{i+1} is larger than y_{i} and y_{i+2} in [n] but it follows y_{i} in E_{i} and precedes y_{i+2} in E_{i+1}, which by the symmetry of U_{x} ⊂ H′ implies that any two y, y′ ∈ U_{x} form a 2-cycle.

Next, |Y ∩ U_{x}| ≤ 1; otherwise pick y_{h}, y_{i} ∈ U_{x} ∩ Y, h < i, with y_{i+1} ∈ Y \ U_{x} and obtain a strictly shorter cycle through (y_{1}, …, y_{h}, y_{i+1}, …, y_{l+1} = y_{1}) as U_{x} ⊂ H′ is ‘symmetric’. The two edges containing the point (if it exists) in Y ∩U_{x} contribute at most 2k−3 to |W |. By the symmetry of U_{x}, we can assume that for the remaining edges E_{i} ∩ U_{x} lies within some fixed (k − 2)-subset of U_{x}, which shows that |W | ≤ 3k − 5.

For k = 2, we can prove a stronger result which includes all infinite families. We exploit the ideas of Kászonyi and Tuza [KT86].

Theorem 6 In the class of cycle-free 2-graphs, we have sat(n, F) = O(n) for any family F.

Proof. It is enough to provide a construction. Repeat the following as long as no forbidden subgraph appears: take the next vertex x of X = [n] and add all of T_{x}. Here, T_{x} is the set of the (oriented) edges of the form xy, y ∈ U_{x}.

Suppose that we have repeated the iteration m = m(n) times. Let G′ = G′(n) be the graph received after these m steps. As [m] ⊂ V (G′) spans the complete cycle-free digraph, the number of iterations is bounded by a constant not depending on n; namely, m < u, where u = min{v(F) : F ∈ F}.

Obviously, m(n) is non-increasing as a function of n for n ≥ u, so it is constant for n sufficiently large. Then, the reason for terminating the procedure is that the addition of T_{m+1} would create a forbidden subgraph F and it will be the case for any subsequent n, that is, G′(n) + T_{m+1} contains the same subgraph F.

Now we add edges to G′ in any order as long as we create neither a cycle nor a forbidden subgraph. In the resulting graph G, no d = |V (F) ∩ U_{m+1}| edges can start at the same vertex y ∈ U_{m}, as otherwise we have a subgraph isomorphic to F. So, the number of edges in G is at most

m(n − 1) − \binom{m}{2} + (n − m)(d − 1) = O(n).

Actually, one can argue that, for sufficiently large n,

m = min{v(F) − α′(F) : F ∈ F} − 1,

where α′(F) is the maximum size of A ⊂ V (F) such that no edge starts in A. Equivalently, m is the minimum number of vertices one needs to remove from some F ∈ F to obtain a directed star (a digraph whose edges start at a common vertex). We can take for d the size of any such star. This observation allows us to write more explicitly the bound of Theorem 6.

4.1.2 Ordered Hypergraphs

We can introduce yet another interesting class: ordered k-graphs. Every ordered k-graph is a usual (unoriented) k-graph with an extra structure: we have a fixed ordering on the vertex set and the vertices of a subgraph inherit their order from the original graph. To avoid a confusion note that an ordered graph comes equipped with a fixed vertex ordering while a cycle-free graph is one that admits at least one compatible vertex ordering.

Without any difficulties we can restate word by word the proof of Theorem 1 (except that now we have already been given an order on the vertex set and in the construction we take the vertices in this order).

Theorem 7 Let F be a family of ordered k-graphs. Suppose that there is s ∈ ℕ such that the following holds for any x ∈ F ∈ F: if U_{x} ⊂ V (F) is an independent set covered by a union of F -edges sharing some vertex y ≤ x, then |U_{x}| ≤ s.

Then we have sat(n, F) = O(n^{k−1}).

Using the ideas of Theorem 6 one can see that, for k = 2, our result can be extended to all infinite families.

Theorem 8 For any family F of ordered 2-graphs, sat(n, F) = O(n).

Trivial examples show that if we enlarge any of the above classes by admitting multiple and/or non-uniform edges, then the estimate (14) fails to be true.

4.2 Layered Hypergraphs

Let t ∈ ℕ be fixed. A layered set X of signature n = (n_{1}, …, n_{t}) (or an nset) is a sequence of t disjoint sets, X = (X_{1}, …, X_{t}) such that |X_{i}| = n_{i}, i ∈ [t]. (Usually we typeset symbols in bold when we want to emphasize that the object has some layered structure.) The components of X are called layers.

Given k = (k_{1}, …, k_{t}), a layered k-graph G is a pair (V (G), E(G)), where V (G) is a layered set and E(G) ⊂ (V (G))^{(k)}, that is, E(G) is a family of k-subsets of V (G). In other words, every k-graph G is a k-graph (usually, given k, we denote k = k_{[t]} = ∑_{i∈[t]} k_{i}, etc.) which comes with a fixed partition of the vertex set into t layers such that every edge intersects the ith layer in exactly k_{i} vertices. The sequence k is called the signature of G; the ith layer of G is denoted by V_{i}(G). For example, a bipartite graph is a layered graph of signature (1, 1) and, for t = 1, we obtain the usual notion of a k-graph. All morphisms between k-graphs preserve layers.

In the obvious way we define the notion of a subgraph, a saturated graph, etc. For example, SAT(n, F) consists of all maximal F-admissible k-graphs on a set of signature n.

It is not very hard to extend Theorem 1 to layered graphs. But, to make this work self-contained, we present a complete proof.

For a k-graph F on X = (X_{1}, …, X_{t}), a set A ⊂ X_{j} is called independent if for every E ∈ E(F), E_{j} ⊄ A.

Theorem 9 If, for a given family F of k-graphs, there exists s such that

1. for every F ∈ F, any independent A ⊂ V_{1}(F) covered by a union of F-edges sharing a vertex in V_{1}(F) \ A, has at most s elements;

2. for every j ∈ [2, t] and F ∈ F, no (s + 1)-set A ⊂ V_{j}(F) can be covered by a set of F-edges coinciding on the first j − 1 layers;

then there exists c = c(F) such that, for any n,

sat(n, F) ≤ c\frac{n_{1}^{k_{1}} × … n_{t}^{k_{t}}}{min(n_{1}, …, n_{t})}.

Proof. As in Theorem 1, we provide a construction of H ∈ SAT(n, F).

Order linearly the vertex set X = (X_{1}, …, X_{t}) so that any vertex of X_{i} comes before any vertex X_{j} for i < j. As usual by U_{x} = {y ∈ X : y > x} we denote the upper shadow of x.

We construct an F-saturated graph H by starting with the empty k-graph H on X and applying the following procedure.

Let j run from 1 to t. Take x ∈ X_{j} in order. For every such x let i vary from 0 to k_{j} −1. Choose one by one C ⊂ X_{j} \U_{x} of size i and let B = C ∪{x}. Given B consider in any order sets A such that A intersects every X_{l} in k_{l} vertices, l ∈ [j − 1], A ∩ X_{j} = B and A ∩ U_{x} = ∅. For every such A we consider T_{A} which is by the definition the family of k-subsets having A as an initial segment. If T_{A} ⊄ E(H) and the addition of the elements of T_{A} to the edge set of H does not create any forbidden subgraph, we add T_{A} to H.

We argue that H exhibits the claimed upper bound in a similar way as in Theorem 1. It is not hard to do, although there are a few new technicalities to overcome.

We define auxiliary k-graphs H_{1}, …, H_{n} on X and auxiliary layered graphs G_{ji} of signature (k_{1}, …, k_{j−1}, i) on the set X_{1} ∪ … ∪ X_{j}, j ∈ [t], i ∈ [k_{j}].

We need these graphs for estimates of e(H) = |E(H)|. H_{x} is the k-graph containing precisely those edges which were added while considering vertices from 1 to x inclusive. The hypergraph G_{ji} contains as edges those A = (A_{1}, …, A_{j}) for which |A_{j}| = i and the set T_{A} was added to H.

We claim that the resulting graph H is F-saturated. Indeed, H is F-admis-sible, as we were adding edges only if it did not produce any forbidden subgraph. On the other hand, take any edge E in the complement of E(H). We did not add E to E(H) when x = max E, j = t, i = k_{t} − 1, A_{l} = E_{l} for l ∈ [t − 1] and A_{t} = E_{t} \ {x} (then T_{A} = {E}). The only reason for this is that it would have created a forbidden subgraph F. Then H + E contains F, which shows that H ∈ SAT(n, F).

We want to show that

e(G_{ji}) ≤ (s − 1)\binom{n_{j}}{i − 1} ∏_{l=1}^{j−1} \binom{n_{l}}{k_{l}},  j ∈ [t], i ∈ [k_{j}].  (15)

(In fact, e(G_{11}) is bounded by some other constant s′ = s′(F) but nothing prevents us from assuming s ≥ s′.) This would establish the theorem as then we would obtain the required

e(H) ≤ ∑_{j=1}^{t} ∑_{i=1}^{k_{j}} e(G_{ji})\binom{n_{j}}{k_{j} − i} ∏_{l=j+1}^{t} \binom{n_{l}}{k_{l}}
≤ ∑_{j=1}^{t} ∑_{i=1}^{k_{j}} ((s − 1)\binom{n_{j}}{i − 1} ∏_{l=1}^{j−1} \binom{n_{l}}{k_{l}}) \binom{n_{j}}{k_{j} − i} ∏_{l=j+1}^{t} \binom{n_{l}}{k_{l}}
= ∑_{j=1}^{t} O (\frac{n_{1}^{k_{1}} × … × n_{t}^{k_{t}}}{n_{j}} ) .

Assume that, for some j and i, estimate (15) is not true. Assume first that i ≠ 1.

For every edge E in G_{ji} consider the set V of its first k_{1} + … + k_{j−1} + i − 1 vertices. When E varies over all edges of G_{ji}, by the pigeon-hole principle some set V appears at least

⌈e(G_{ji}) (\binom{n_{j}}{i − 1} ∏_{l=1}^{j−1} \binom{n_{l}}{k_{l}})^{−1}⌉ ≥ s

times. Let V consist of classes V_{1}, …, V_{j} of sizes k_{1}, …, k_{j−1}, i − 1 respectively.

Let E_{1}, …, E_{s} ∈ E(G_{ji}) be s distinct edges of G_{ji} containing V as an initial segment, say E_{l} = V ∪ {z_{l}}, l = 1, …, s, V < z_{1} < … < z_{s}. Let z = max V.

Since E_{1} ∈ E(G_{ji}), all edges whose initial segment is E_{1} were added to H at the moment when x = z_{1}, A = V ∪ {z_{1}}. It follows that V ∉ E(G_{j,i−1}), for otherwise these edges would have already been present in H. The only reason that we did not add V to E(G_{ji}) earlier, when x = z, C = V_{j} \ {x} and A = V, must have been that the k-graph H′ = H_{x} + T_{V} contains some forbidden subgraph F ∈ F. Let

A = {u ∈ X_{j} ∩ U_{z} : u ∈ E for some E ∈ T_{V} ∩ E(F)}.  (16)

By Assumption 1 (for j = 1) or by Assumption 2 (for j ≥ 2) of the theorem, |A| ≤ s. One can argue that any layer-preserving permutation σ of X affecting only U_{z} is an automorphism of H_{z}, because any T_{B} with z ≥ max B is σinvariant. Therefore, we may assume that A ⊂ Z = {z_{1}, …, z_{s}}.

Now let E ∈ E(F) \ E(H) ⊂ T_{V}. Clearly, E ∈ T_{E_{l}} ⊂ E(H), where z_{l} = min(E ∩ {z_{1}, …, z_{s}}), since E_{l} ∈ E(G_{ji}); the obtained contradiction F ⊂ H proves (15) for j ∈ [t], i ∈ [2, k_{j}].

Suppose that (15) is not true for i = 1. Then as before we argue that there are at least s edges in G_{j1}, say V_{1}, …, V_{s} ∈ E(G_{j1}), such that their restrictions to X_{1} ∪ … ∪ X_{j−1} are the same which we denote by V. Let V_{l} ∩ X_{j} = {v_{l}}, l ∈ [s].

First, if j > 1 then as above we argue that V is not in G_{j−1,k_{j−1}} because T_{V_{1}} ⊂ T_{V} was added later. The only reason for omitting V is that the addition of T_{V} would have created a forbidden F. The set A defined by (16) has at most s elements by Assumption 2; we can assume that A ⊂ {v_{1}, …, v_{s}} and deduce a contradiction.

Finally, if j = 1 then H contains all k-edges E intersecting {v_{1}, …, v_{s}} and s = O(1) follows.

A version for bipartite graphs (that is, (1, 1)-graphs) covers all (including infinite) families and uses slightly different ideas.

Theorem 10 For any family F of bipartite graphs, there is c = c(F) such that, for any n_{1}, n_{2} > 0,

sat(n_{1}, n_{2}, F) ≤ c\frac{n_{1}n_{2}}{min(n_{1}, n_{2})}.

Proof. Suppose first that n_{1} ≥ n_{2}. Choose a large s = s(F) (to be specified later). If n_{2} < s then any (n_{1}, n_{2})-bipartite graph contains O(n_{1}) vertices and we are home. Otherwise, as long as no forbidden subgraph appears, take one by one vertices in the first layer and for every such vertex x ∈ X_{1} = [n_{1}] add all edges connecting it to X_{2} to obtain a graph H′. Suppose we do it m times. Note that as n_{2} → ∞ then m = m(n_{2}) does not increase so we can assume that m is constant for every n_{2} ≥ s, some s = s(F). Then, the only thing preventing us from adding the edges {{m + 1, y} : y ∈ X_{2}} is the creation of a forbidden subgraph F. Let |V (F) ∩ X_{2}| = l. We see that if we draw through any point x ∈ X_{1} \ [m] any l edges, we would obtain a copy of F. Therefore, in whatever way we complete H′ to H ∈ SAT(n_{1}, n_{2}, F), we would have

e(H) ≤ mn_{2} + ln_{1} ≤ (l + m)n_{1} = O(n_{1}).

We settle the case n_{1} ≤ n_{2} in the same manner.

4.3 Forbidden Matrices

Here we investigate sat-type problems for 01-matrices. We show that sat(n, F) = O(n^{k−1}) for any family F of k-row matrices and indicate other results.

The expression ‘n × m-matrix’ means a matrix with n rows (which we view as horizontal arrays) and m vertical columns. We restrict entries to only two values, 0 and 1. For an n × m-matrix M, its order v(M) = n is the number of rows and its size e(M) = m is the number of columns. Please distinguish expressions like ‘an n-row matrix’ and ‘an n-row’ standing respectively for a matrix with n rows and for a row containing n elements.

A matrix F is a submatrix of a matrix A (denoted F ⊂ A) if deleting some set of rows and columns of A we can obtain a matrix which is a row/column permutation of F. Given a family F of matrices (referred to as forbidden), we say that a matrix M is F-admissible (or F-free) if M contains no F ∈ F as a submatrix. A simple matrix M (that is, a matrix without repeated columns) is called F-saturated if M is F-admissible but the addition of any column not present in M violates this property; this is denoted by M ∈ SAT(n, F), n = v(M). Please note that, although the definition requires that M is simple, we allow multiple columns in matrices belonging to F.

A popular extremal problem is to consider forb(n, F), the maximum size of a simple F-admissible matrix with n rows or, equivalently, the maximal size of M ∈ SAT(n, F). For example, the fundamental formula (17) falls into this class. The interested reader may start with a recent paper by Anstee, Griggs and Sali [AGS97] containing many references.

On the other hand, the ‘dual’ of the forb-type problem has received little attention so far. Namely, one can ask what is the value of sat(n, F), the minimal size of an F-saturated matrix with n rows:

sat(n, F) = min{e(M) : M ∈ SAT(n, F)}.

We will be mainly interested in this function. Obviously, sat(n, F) ≤ forb(n, F). If F = {F } consists of a single forbidden matrix F then we write SAT(n, F) = SAT(n, {F }), etc.

For an n × m-matrix M and sets A ⊂ [n] and B ⊂ [m], M(A, B) denotes the corresponding |A|×|B|-submatrix of M. We use the following self-obvious shorthands: M(A,) = M(A, [m]), M(A, i) = M(A, {i}), etc. For example, the rows and the columns of M are denoted by M(1,), …, M(n,) and M(, 1), …, M(, m) respectively while individual entries—by M(i, j), i ∈ [n], j ∈ [m].

The n×(m_{1} +m_{2})-matrix [M_{1}, M_{2}] is obtained by concatenating an n×m_{1}- matrix M_{1} and an n × m_{2}-matrix M_{2}. Let mM = [M, …, M] denote m copies of M. We write N ≅ M to say that N is a column/row permutation of M. Thus, N ⊂ M if N ≅ M(A, B) for some index sets A and B.

By T_{k}^{l} we denote the simple k × \binom{k}{l}-matrix consisting of all k-columns with exactly l ones and by K_{k}—the k × 2^{k} matrix of all possible columns of size k. Naturally, T_{k}^{≤l} denotes the k × \binom{k}{≤l}-matrix consisting of all distinct columns with at most l ones, etc. (We use the shortcut \binom{k}{≤l} = \binom{k}{0} + \binom{k}{1} + … + \binom{k}{l}.)

We will need the following result proved independently by Vapnik and Chervonenkis [VC71], Perles and Shelah (see [She72]) and Sauer [Sau73].

forb(n, K_{k}) = \binom{n}{≤ k − 1} = ∑_{i=0}^{k−1} \binom{n}{i}.  (17)

Suppose that F consists of k-row matrices. Is there any good general upper bound on forb(n, F) or sat(n, F)? There were different papers dealing with general upper bounds on forb(n, F), e.g. by Anstee and Füredi [AF86], by Frankl, Füredi and Pach [FFP87] and by Anstee [Ans95], until the conjecture of Anstee and Füredi [AF86] that forb(m, F) = O(n^{k}) for any fixed F was elegantly proved by Füredi (see [AGS97] for a proof).

On the other hand, we can show that sat(n, F) = O(n^{k−1}) for any family F of k-row matrices (including infinite families). Note that we cannot decrease the exponent of k − 1 with the estimate remaining true for any F; for example, sat(n, T_{k}^{k}) = \binom{n}{≤k−1} as T_{n}^{<k} is the only matrix in SAT(n, T_{k}^{k}).

Theorem 11 For any family F of k-row matrices, sat(n, F) = O(n^{k−1}).

Proof. We may assume that K_{k} is F-admissible for otherwise we are home by (17) as then sat(n, F) ≤ forb(n, K_{k}) = O(n^{k−1}).

Let l ∈ [0, k] be the smallest number such that there exists m for which [mT_{k}^{≤l}, T_{k}^{>l}] is not F-admissible. Clearly, l is well-defined as, for l = k, we obtain the matrix mK_{k} which, of course, is not F-admissible for large m.

Let d ≥ 1 be the maximal integer such that [mT_{k}^{<l}, dT_{k}^{l}, T_{k}^{>l}] is F-admissible for any m. Observe that letting d equal 1 we obtain the matrix [mT_{k}^{<l}, T_{k}^{≥l}] which is F-admissible. Indeed, for l > 0 this is true by the choice of l; for l = 0 we have K_{k} which is F-admissible by our assumption. By the choice of l, d is bounded, that is, d is well-defined.

Choose any m such that [mT_{k}^{<l}, (d + 1)T_{k}^{l}, T_{k}^{>l}] is not F-admissible.

Suppose first that l < k. Given n, let N ⊂ T_{n}^{l+1} be the n-row matrix corresponding to the following set system:

H = ⋃_{j∈[d]}{Y ∈ [n]^{(l+1)} : ∑_{y∈Y} y ≡ j (mod n)}.

Note that any A ∈ [n]^{(l)} is covered by at most d edges of H as there are at most d possibilities to choose i ∈ [n] \ A so that A ∪ {i} ∈ H: i ≡ j − ∑_{a∈A} a (mod n), j ∈ [d].

On the other hand, the set H_{1} of all l-subsets of [n] covered by fewer than d edges of H has size at most 2d\binom{n}{l−1}. Indeed, if A ∈ H_{1} then, for some j ∈ [d] and x ∈ A, 2x = j −∑_{a∈A−x} a (mod n) so, once A\{x} and j have been chosen, there are at most 2 choices for x.

Call X ∈ [n]^{(k)} bad if, for some A ∈ X^{(l)},

|{Y ∈ H : Y ⊃ A, Y ∩ (X \ A) = ∅}| ≤ d − 1.  (18)

To obtain a bad k-set X, we either complete some A ∈ H_{1} to any k-set or take any l-set A and let X ⊃ A intersect some H-edge covering A. Therefore, the number of bad sets is at most

2d\binom{n}{l − 1}\binom{n}{k − l} + \binom{n}{l}d\binom{n}{k − l − 1} = O(n^{k−1}).

Assume that n is so large that N(X,) ⊃ mT_{k}^{<l} for any X ∈ [n]^{(k)}. This is possible as d ≥ 1. Of course, e(N) = O(n^{k−1}).

Clearly, N(X,) ⊂ [d\binom{n}{l}T_{k}^{<l}, dT_{k}^{l}, T_{k}^{l+1}], for any X ∈ [n]^{(k)}. Hence, N cannot contain a forbidden submatrix by the choice of l and d. Now complete it to an arbitrary M = [N, N_{1}] ∈ SAT(n, F).

Suppose that e(N_{1}) ≠ O(n^{k−1}). Then, by (17), K_{k} ≅ N_{1}(X, Y) for some X, Y. Now, remove the columns corresponding to Y from N_{1} and repeat the procedure as long as possible to obtain more than O(n^{k−1}) column-disjoint copies of K_{k} in N_{1}. If some X ∈ [n]^{(k)} appears more than d times, then M(X,) ⊃ [mT_{k}^{<l}, (d + 1)K_{k}] is not F-admissible. Otherwise, K_{k} ⊂ N_{1}(X,) for some good (ie. not bad) X ∈ [n]^{(k)}; but then N(X,) ⊃ dT_{k}^{l} and

M(X,) ⊃ [mT_{k}^{<l}, dT_{k}^{l}, K_{k}]

contains a forbidden matrix. This contradiction proves the required bound for l < k.

Let us consider the case when l = l(F) equals k; the above argument does not work in this case because N has size Θ(n^{k}), which is too large.

Consider the family H obtained by interchanging zeros and ones in each F ∈ F. Clearly, sat(n, H) = sat(n, F). If l(H) < k, then we are home by the above argument applied to H. So, we assume that l(H) = k.

Consider first the case k = 1. Let F ∈ F be a matrix of the smallest size f. Let the only row of F consist of f_{0} zeros and f_{1} ones; f_{0} + f_{1} = f. Note that f_{1} ≥ 2 and f_{0} ≥ 2, because l(F) = l(H) = 1. Trivially, for any n there exists a simple n × (f − 1)-matrix M such each row of M contains exactly f_{0} zeros. By the minimality of f, M is F-admissible. When we try to complete M to any F-saturated matrix, any added column cannot contain an entry equal to 1; hence, all we can add is at most one all-zero column. Hence, sat(n, F) ≤ f for any n, which implies the required.

So assume that k ≥ 2. Now we repeat a part of the above proof with some modifications. Probably, it would be possible to write a general single argument covering all the cases, but we are afraid that the proof would be very hard to follow then.

Let l′ ∈ [0, k − 1] be the smallest number such that there exists m for which [mT_{k}^{≤l′}, T_{k}^{>l′}, T_{k}^{k−1}, mT_{k}^{k}] is not F-admissible. Observe that l′ is well-defined as this matrix contains mK_{k} as a submatrix if we let l′ = k − 1.

Define d to be the maximal integer such that [mT_{k}^{<l′}, dT_{k}^{l′}, T_{k}^{>l′}, T_{k}^{k−1}, mT_{k}^{k}] is F-admissible for any m. Note that letting d = 1 we obtain the matrix [mT_{k}^{<l′}, T_{k}^{≥l′}, T_{k}^{k−1}, mT_{k}^{k}] which does not contain a forbidden submatrix. Indeed, if l′ > 0, this is true by the choice of l′; if l′ = 0, then our matrix [K_{k}, T_{k}^{k−1}, mT_{k}^{k}] is necessarily F-admissible as l(H) = k > 1 by our assumption.

Choose any m such that [mT_{k}^{<l′}, (d + 1)T_{k}^{l′}, T_{k}^{>l′}, T_{k}^{k−1}, mT_{k}^{k}] is not F-free.

Let N be the n-row matrix corresponding to the following set system:

H = ⋃_{j∈[d]}{Y ∈ [n]^{(l′+1)} : ∑_{y∈Y} y ≡ j (mod n)}.

As above we observe that every A ∈ [n]^{(l′)} is covered by at most d edges of H and the number of bad sets (that is, such X ∈ [n]^{(k)} that (18) holds for some A ∈ X^{(l′)}) is O(n^{k−1}). Assume that n is so large that N(X,) ⊃ mT_{k}^{<l′} for any X ∈ [n]^{(k)}, which is possible as d ≥ 1.

Let M_{1} = [N, T_{n}^{≥n−1}]. Clearly,

M_{1}(X,) ⊂ [d\binom{n}{l′}T_{k}^{<l′}, dT_{k}^{l′}, T_{k}^{l′+1}, T_{k}^{k−1}, nT_{k}^{k}], for any X ∈ [n]^{(k)}.

Hence, M_{1} cannot contain a forbidden submatrix by the choice of l′ and d. Now complete it to an arbitrary M = [M_{1}, M_{2}] ∈ SAT(n, F).

Clearly, e(M_{1}) = O(n^{k−1}). Suppose that e(M_{2}) ≠ O(n^{k−1}). Then, by (17), K_{k} ≅ M_{2}(X, Y) for some X, Y. Now, remove the columns corresponding to Y from M_{2} and repeat the procedure as long as possible to obtain more than O(n^{k−1}) column-disjoint copies of K_{k} in M_{2}. If some X ∈ [n]^{(k)} appears more than d times then M(X,) ⊃ [mT_{k}^{<l′}, (d + 1)K_{k}, T_{k}^{k−1}, mT_{k}^{k}] is not F-admissible. (We assume n ≥ m + k.) Otherwise, K_{k} ⊂ M_{2}(X,) for some good (ie. not bad) X ∈ [n]^{(k)}; but then N(X,) ⊃ dT_{k}^{l′} and M(X,) ⊃ [mT_{k}^{<l′}, dT_{k}^{l′}, T_{k}^{k−1}, mT_{k}^{k}, K_{k}] contains a forbidden matrix. This contradiction proves the theorem.

Let us present some other results.

The following simple observation is useful in tackling sat-type problems. Suppose that no forbidden matrix has two equal rows. Let M′ be obtained from M ∈ SAT(n, F) by duplicating the nth row of M, that is, we let M′([n],) = M and M′(n + 1,) = M(n,). Complete M′, in an arbitrary way, to an Fsaturated matrix. Let C be any added (n + 1)-column. As both M′([n],) and M′([n − 1] ∪ {n + 1},) are equal to M ∈ SAT(n, F), we conclude that both C([n]) and C([n − 1] ∪ {n + 1}) must be columns of M. As C is not an M′- column, C = (C′, b, 1 − b) for some (n − 1)-column C′ such that both (C′, 0) and (C′, 1) are columns of M. This implies that sat(n + 1, F) ≤ e(M) + 2l, where l is the number of pairs of equal columns in M after we delete the nth row. In particular, the following theorem follows.

Theorem 12 Suppose that no matrix in F has two equal rows. Then either sat(n, F) is constant for large n or sat(n, F) ≥ n + 1 for every n.

Proof. If we have some M ∈ SAT(n, F) with at most n columns then a wellknown theorem of Bondy [Bon72a] (see e.g. [Bol86, Theorem 2.1]) implies that there is i ∈ [n] such that the removal of the ith row does not produce multiple columns. Now the duplication of the ith row gives an F-saturated matrix, which implies sat(n + 1, F) ≤ sat(n, F), and the theorem follows.

There are many open problems concerning particular forbidden matrices; for example, the computation of sat(n, T_{m}^{k}) or sat(n, K_{k}). Of course, Theorem 12 is applicable here. While it is easy to see that sat(n, T_{m}^{k}) ≥ n+1 for any m ∈ [0, k] and k ≥ 2, we do not know for which k we have sat(n, K_{k}) = O(1). We could only show that sat(n, K_{2}) = n+1, which is an easy (and perhaps known) result, and (surprisingly) sat(n, K_{3}) = 10 for n ≥ 4. We do not provide any proofs here, except we exhibit an example of an n-row K_{3}-saturated matrix of size 10 for any n ≥ 6. For n = 6 we can take

M = [ \begin{matrix}0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \end{matrix} ] .

It is possible (but rather boring) to check by hand that M is indeed K_{3}-saturated as is, in fact, any n × 10-matrix M′ obtained from M by duplicating any row, cf. Theorem 12. (The symmetries of M shorten the verification.)

The author would like to thank Richard Anstee for drawing the author’s attention to the Turán-type problem for matrices and Andrew Thomason for his computer programme used for computing sat(n, F) for small n and F.

4.4 Edge Killers

Here we introduce certain extremal problems which are closely related to the sat-type questions. We settle the problem for complete 2-graphs, which extends a theorem of Erdős, Hajnal and Moon [EHM64] who computed sat(n, K_{m}^{2}).

Given a forbidden family F, we say that a k-graph G F-kills (or simply kills when F is understood) an edge E ∈ E(\overline{G}) if the addition of E to G creates a new forbidden subgraph. For example, G ∈ m-SAT(n, F) if and only if it kills all edges in its complement. The F-closure Cl_{F}^{∗}(G) of G is the k-graph on V (G) consisting of all edges of G plus all F-killed edges. Let cl_{F}^{∗}(G) = |Cl_{F}^{∗}(G)|.

Let us define k-m-sat(e, F; n) to be the maximum size of Cl_{F}^{∗}(G) where G is a k-graph of order n and size e, e ≤ m-sat(n, F). In the same way we define k-sat(e, F; n) except we consider only F-free graphs of order n and size e, e ≤ sat(n, F). We agree that k-sat = k-m-sat = \binom{n}{k} for other (larger) values of e. Clearly, k-m-sat(e, F; n) ≥ k-sat(e, F; n); both k-m-sat and k-sat are monotone increasing in e.

Here we compute k-m-sat and k-sat (and describe all extremal graphs) for complete 2-graphs. This extends a result of Erdős, Hajnal and Moon [EHM64] who computed sat(n, K_{m}^{2}), as sat(n, F) = min\{e : k-sat(e, F; n) = \binom{n}{k}\}.

Let us provide a construction. Given n ≥ m ≥ 3 and e,

\binom{m}{2} − 1 ≤ e ≤ \binom{n}{2} − \binom{n − m + 2}{2} = m-sat(n, K_{m}^{2}) = sat(n, K_{m}^{2}),

write e − \binom{m−2}{2} = l(m − 2) + r with r ∈ [0, m − 3]. Choose an (m − 2)-set A and a disjoint l-set B. Let G be P_{A,B} (which consists of all edges lying within A ∪ B and intersecting A) plus any r extra edges, none within B. (So B is an independent set in G.) It is routine to check that G can be accommodated within [n]. Clearly, G kills all \binom{l}{2} edges of K^{2}(B). We show that this is best possible by applying the contraction technique of Erdős, Hajnal and Moon [EHM64].

Theorem 13 In the above notation,

k-sat(e, K_{m}^{2}; n) = k-m-sat(e, K_{m}^{2}; n) = \binom{l}{2} + e,  (19)

and all extremal graphs are given by the construction preceding the theorem.

Proof. To prove the upper bound, we use induction on l with the case l = 2 being trivially true. Let l ≥ 3. Given a graph G of order n and size e (not necessarily K_{m}^{2}-free), fix any killed edge {x_{1}, x_{2}} and let G′ be obtained from G by contracting the vertices x_{1} and x_{2} into one vertex x. Fix an (m − 2)-set Y such that G[Y ∪ {x_{1}, x_{2}}] is the complete graph but for {x_{1}, x_{2}}; colour these \binom{m}{2} − 1 edges red. Clearly, during the contraction at least m − 2 red edges disappear, so e(G′) ≤ e(G) − m + 2.

Obviously, an edge killed by G is also killed by G′ (except {x_{1}, x_{2}}) but two G-killed edges, say \{a, x_{1}\}, \{a, x_{2}\} ∈ E(\overline{G}), may produce only one edge in G′ (which is also killed). When this happens then, for i = 1, 2, choose an arbitrary (m − 2)-set X_{i} with G[X_{i} ∪ {a, x_{i}}] = P_{m−2,2} and colour all edges connecting a to X_{i} blue. Let D be a blue edge. Some a ∈ D is incident neither to x_{1} nor to x_{2}, so D is not coloured red. As the other endvertex of D sends at least one edge to {x_{1}, x_{2}}, D cannot be coloured blue more than twice.

We have e − \binom{m}{2} + 1 non-red edges each being coloured blue at most twice, while each time two killed edges contract together exactly 2(m − 2) edges are coloured blue. This yields

cl_{K_{m}^{2}}^{∗}(G) − e(G) ≤ cl_{K_{m}^{2}}^{∗}(G′) − e(G′) + ⌊\frac{e − \binom{m}{2} + 1}{m − 2}⌋ + 1.  (20)

(the last term 1 counts the edge {x_{1}, x_{2}}) and the induction assumption applied to the graph G′ of order n − 1 and size at most e − m + 2 proves the desired upper bound.

Let us follow our argument to characterize the cases of equality. Clearly, for l = 2, when e = \binom{m}{2} − 1 + r, we must have an induced P_{m−2,2}-subgraph present while the remaining r edges can be placed arbitrarily, which is precisely what our construction says.

Let l ≥ 3 and let G be an extremal graph. Apply the above contraction to G, preserving the above notation. By induction, G′ = P_{A,B} + E_{1} + … + E_{r}, where B is an independent (l −1)-set disjoint from an (m−2)-set A. The vertex x, which has degree at least m − 2 in G′, must belong to A ∪ B as r ≤ m − 3.

Suppose that x ∈ A. Then the (m−1)-set Y ∪{x}, which spans the complete graph in G′, must equal A∪{y}, for some y ∈ B. Each blue edge of G lies within a P_{m−2,2}-subgraph in G′ and, as r ≤ m − 3, none of E_{1}, …, E_{r} can be blue (nor red, of course). But then, for z ∈ B \{y}, E(G) contains either {x_{1}, z} or {x_{2}, z} (because {x, z} ∈ E(G′)) which is also uncoloured. So, we have at least r + 1 uncoloured edges and we cannot have equality in (20), which is a contradiction.

Hence, x ∈ B; then Y must equal A, and G is given by our construction.

5 Irregularities

Here we demonstrate many irregularities of the sat-function in the comparison to the Turán function ex(n, F) = max{e(G) : G ∈ SAT(n, F)}.

Clearly, ex(n, F_{1}) ≤ ex(n, F_{2}) whenever F_{1} is a subgraph of F_{2}. Kászonyi and Tuza [KT86] demonstrated an example of F_{1} ⊂ F_{2} with sat(n, F_{1}) > sat(n, F_{2}) for all large n. Tuza [Tuz92, p. 401] asks if there exists a connected irregular pair F_{1} ⊂ F_{2}; this is answered in the affirmative by the following simple example.

Example 14 There is a pair of connected graphs F_{1} ⊂ F_{2} on the same vertex set such that sat(n, F_{1}) > sat(n, F_{2}) for all n ≥ v(F_{1}).

Proof. Let m ≥ 5 and F_{1} = S_{m}^{2}, that is, V (F_{1}) = [m] and E(F_{1}) = {{1, i} : i ∈ [2, m]} and let F_{2} be obtained from F_{1} by adding the edge {2, 3}. Clearly, sat(n, F_{2}) ≤ n − 1, n ≥ m, as S_{n}^{2} is an example of an F_{2}-saturated graph.

On the other hand, in any monotonically F_{1}-saturated graph G, any two vertices of degree at most m − 3 must be connected. (Otherwise the addition of this edge cannot create a forbidden subgraph.) If we have v ∈ [0, m − 2] such vertices, then e(G) ≥ \binom{v}{2}+(m−2)(n−v)/2, which is easily seen to exceed n−1 for all n ≥ m.

Remark. Curiously enough, the w-sat-function (studied later) exhibits the analogous irregularity on the very same pair: it is not hard to check that w-sat(n, F_{2}) = e(F_{2}) − 1 = m − 2 while w-sat(n, F_{1}) = \binom{m−1}{2}, n ≥ m.

Clearly, for every n ≥ v(F), we have ex(n, F) ≤ ex(n + 1, F). On the other hand, Kászonyi and Tuza [KT86] observe that, for any n = 2k − 1, sat(n, P_{3}) = k +1 > sat(n+1, P_{3}) = k, where P_{3} is the path with 3 edges. Our next example amplifies this irregularity.

Example 15 For every constant d, there is a 2-graph F = F (d) such that

sat(n, F) < sat(n ± 1, F) − d,

for a periodic series of values of n.

Proof. Let m = 2d + 3 and let F = B_{mm} be the dumb-bell

E(B_{mm}) = [m]^{(2)} ∪ [m + 1, 2m]^{(2)} ∪ {{1, m + 1}},

that is, B_{mm} is the disjoint union of two copies of K_{m} plus one edge connecting them.

Let us show that the claim is true for any n = lm if l ∈ ℕ is large. Clearly, sat(lm, F) ≤ lm(m − 1)/2 (in fact, this is sharp) as lK_{m}^{2} ∈ SAT(lm, F). On the other hand, let n = lm − 1 and suppose that G ∈ sat(n, F) has at most g = lm(m − 1)/2 + d edges.

Clearly, δ(G), the minimal degree of G, is at least δ(B_{mm}) − 1 = m − 2. Suppose that for some x ∈ V (G) d(x) = m − 2. Then for every y non-incident to x the edge E = \{x, y\} ∈ E(\overline{G}) cannot be the bridge in a created B_{mm}- subgraph as the degree of x is too small; that is, x and y fall in the same K_{m}^{2}-half. Therefore, y must be connected to all m − 2 neighbours of x and e(G) ≥ (m − 2)n + O(1) which is a contradiction.

Hence δ(G) ≥ m − 1. The inequality Δ(G) + (m − 1)(n − 1) ≤ 2e(G) ≤ 2g implies that Δ(G) ≤ 2(d + m − 1). If some x ∈ V (G) does not belong to an m-clique then any missing edge {x, y} must create a K_{m}^{2}-subgraph and we arrive at a contradiction again, as d(x) ≤ Δ(G) is bounded. Thus the whole of V (G) is covered by m-cliques.

We want to find a set X ⊂ V (G) with the surplus s(X) = e(G[X])− \frac{m−1}{2}|X| at least m − 1 as then the claim would follow:

e(G) ≥ e(G[X]) + \frac{m − 1}{2}(n − |X|) ≥ \frac{m − 1}{2}n + m − 1 > g.

As m does not divide n, there are two distinct cliques A, B ∈ V (G)^{(m)} with i = |A ∩ B| > 0. It is straightforward to verify that

s(A ∪ B) = 2\binom{m}{2} − \binom{i}{2} − \frac{m − 1}{2}(2m − i) ≥ \frac{m − 1}{2}.

No m-clique C ⊄ A∪B can intersect some other clique or A∪B. (Otherwise we gain another suplus of (m − 1)/2.) By the divisibility argument, i = 1. As a (2m − 1)-clique has suplus at least m − 1, there exists some E ∈ E(\overline{G}) lying within A∪B. It is easy to see that G+E must contain a K_{m}^{2}-subgraph on some m-set C ⊄ A ∪ B intersecting A ∪ B, which implies s(A ∪ B ∪ C) ≥ m − 1 as required.

Clearly, for n = ml + 1, sat(n, B_{mm}) ≥ \frac{m−1}{2}n > g, which completes the proof.

The elegant averaging argument of Katona, Nemetz and Simonovits [KNS64] shows that the limit ex(n, F)/n^{k} exists for any family F of k-graphs. Concerning the sat-function, Tuza [Tuz88] made the following (still open) conjecture.

Conjecture 16 (Tuza) For any 2-graph F, the limit lim_{n→∞} sat(n, F)/n exists.

We can show that this assertion is not true for families of forbidden graphs.

Example 17 There exists a finite family F of 2-graphs such that, for some c > 0 and for infinitely many n, sat(n, F) < sat(n ± 1, F) − cn. In particular, the ratio sat(n, F)/n does not necessarily tend to a limit for a finite family F of 2-graphs.

Proof. Fix m ≥ 4 and consider the family F consisting of the dumb-bell B_{mm} and F_{m1}, …, F_{m,m−1}, where

E(F_{mi}) = [m]^{(2)} ∪ [m − i + 1, 2m − i]^{(2)}, i ∈ [m − 1],

that is, F_{mi} is the union of two K_{m}^{2}-graphs sharing i common vertices.

Clearly, the disjoint union of K_{m}^{2}-graphs is F-saturated as any missing edge connects two different copies and thus creates a B_{mm}-subgraph. Hence, if m divides n then sat(n, F) ≤ \frac{n}{m} \binom{m}{2}.

On the other hand, suppose that m does not divide n and let G be any F-saturated graph on [n]. By the definition of F, no vertex can belong to two different K_{m}^{2}-subgraphs of G; suppose that the sets A_{i} = [m(i − 1) + 1, mi], i ∈ [s], are all m-sets spanning complete subgraphs in G.

Note the following two properties of G. Property A: G[A_{[s]}] ≅ sK_{m}^{2}. (Because B_{mm} is forbidden.) Property B: any missing edge E intersecting B = [n] \ A_{[s]} creates a K_{m}^{2}-subgraph. (Because it is impossible that B_{mm} ⊂ G + E with E being the bridge.)

We claim that these two properties and the fact that B ≠ ∅ (as m is not a divisor of n) imply that

e(G) ≥ \frac{n}{m} (\binom{m}{2} + m − 2) − m^{2}.  (21)

We use induction on n. If some E ∈ B^{(2)} is not a G-edge then it is easy to check that the graph G′ obtained from G by contracting the edge E has the properties in question. The endvertices of E have at least m − 2 common neighbours in G (because E creates a K_{m}^{2}-subgraph) so e(G) ≥ e(G′) + m − 2 and (21) follows by induction.

Suppose that B spans the complete graph in G. If some E ∈ E(\overline{G}) intersects both A_{i} and B then a K_{m}^{2}-subgraph created by E lies within A_{i} ∪ B and so at least m − 2 G-edges intersect both A_{i} and B. Therefore,

e(G) ≥ f(b) = (n − b)\frac{m − 1}{2} + \binom{b}{2} + \frac{n − b}{m}(m − 2),

where b = |B|. (We correspondingly count the edges within A_{[s]}, within B and in between.) The minimum of f is achieved for b = \frac{m}{2} + \frac{m−2}{m} and our estimate (21) follows rather crudely.

Hence, if we increase/decrease n = ml by one, then sat(n, F) increases at least by n\frac{m−2}{m} + O(1).

6 Specific Classes

Our aim in this section is to give precise information about sat(n, F) for special classes F.

6.1 Stars

The star S_{m}^{k} = P (1, m − 1; 1, k − 1), m > k ≥ 2, has [m] as the vertex set and {E ∈ [m]^{(k)} : E ∋ m} as the edge set. In other words, S_{m}^{k} has m vertices and its \binom{m−1}{k−1} edges are the k-tuples containing some fixed vertex, which is called the centre.

The exact values of sat(n, S_{m}^{k}) are known only for S_{m}^{2}, any m, (see [KT86]) and for S_{4}^{3} (see [EFT91]).

The asymptotic behaviour of sat(n, S_{k+1}^{k}) was found by Erdős, Füredi and Tuza [EFT91, Theorem 2]. Exploiting their ideas we extend their result to all stars; this theorem appears in [Pik99b].

Theorem 18 Let m > k ≥ 2 and S = S_{m}^{k}. Then

\frac{m − k}{2} \binom{n}{k − 1} ≥ sat(n, S) ≥ m-sat(n, S) ≥ \frac{m − k}{2} \binom{n}{k − 1} − O(n^{k−4/3}). (22)

Proof. Let us provide a construction of an S-saturated graph G = G_{m,n}^{k} of order n proving the upper bound. Partition the vertex set [n] into n′ = ⌈n/(m−k+1)⌉ blocks B_{1}, …, B_{n′} of size m − k + 1 each except possibly the last one. The edge set is

E(G) = {F ∈ [n]^{(k)} : |F ∩ B_{j}| ≥ 2, j = min{i ∈ [n′] : F ∩ B_{i} ≠ ∅}} .

Thus every edge of G has at least two common points with some B_{j} and intersects no B_{i} with i < j.

Let us show that S ⊄ G. Suppose not and we have an S-subgraph S′ ⊂ G centered at x. Let

j = min{i ∈ [n′] : V (S′) ∩ B_{i} ≠ ∅}.  (23)

Choose a k-set F ∋ x so that it contains one vertex from B_{j} and some k − 1 vertices in V (S′) \ B_{j} which is possible since |V (S′) \ B_{j}| ≥ k − 1. We obtain a contradiction as on one hand F contains the centre x and must belong to S while on the other hand F ∉ E(G) by definition.

If we add any extra edge F to G then the set Y = F ∪ B_{j} spans a copy of S centered at x where B_{j} is the first block intersecting F and {x} = F ∩ B_{j}. Indeed, every F′ ∈ Y^{(k)} containing x either equals F or intersects B_{j} in at least two points and so belongs to E(G).

Therefore we conclude that G is S-saturated. To prove the desired upper bound |G_{m,n}^{k}| ≤ \frac{m−k}{2} \binom{n}{k−1} we observe, for k = 2, that each vertex of the 2graph G_{m,n}^{2} has degree at most m − k while, for k ≥ 3, we use induction and the equality |G_{m,n+1}^{k}| = |G_{m,n}^{k}| + |G_{m−1,n}^{k−1}|.

Of course, sat(n, S) ≥ m-sat(n, S).

Finally, let G be a minimum monotonically S-saturated graph on V = [n]. By the definition, the addition to G of any edge F ∈ E(\overline{G}) creates at least one S-subgraph S′ ⊂ G + F. Let S(F) be the set of all such subgraphs S′ created by F.

Let F(F) denote the set of edges in \overline{G} which intersect F ∈ E(G) in k − 1 points and create a copy of S containing F as an edge. Formally,

F(F) = \{F′ ∈ E(\overline{G}) : |F ∩ F′| = k − 1, ∃S′ ∈ S(F′) F ∈ E(S′)\} , F ∈ E(G).

Also we define

\begin{aligned}F(G′) & = & ⋃_{F ∈E(G′)} F(F), & G′ ⊂ G, \\ ∂F & = & F^{(k−1)}, & F ∈ [n]^{(k)}, \\ ∂G′ & = & ⋃_{F ∈E(G′)} ∂F, & a k-graph G′. \end{aligned}

As G is monotonically S-saturated we conclude that

F(G) = V^{(k)} \ E(G).  (24)

Choose an integer t = t(n), to be specified later, such that t → ∞ and t/n → 0. On the vertex set V we define two subgraphs G_{0}, G_{1} ⊂ G; G_{0} is a maximal subgraph of G with |F(G_{0})| ≤ t|G_{0}| and G_{1} consists of the edges of G not in G_{0}: E(G_{1}) = E(G) \ E(G_{0}). By the maximality of G_{0} for every F ∈ E(G_{1}) we have

|F(F) \ F(G_{0})| > t.  (25)

From (24) and the proved upper bound in (22) we conclude that |F(G)| = \binom{n}{k}−|G| = \binom{n}{k}−O(n^{k−1}). Taking into the account that F(G) = F(G_{0})∪F(G_{1}) and |F(G_{0})| ≤ t|G_{0}| = O(tn^{k−1}) we obtain

|X| = \binom{n}{k} − O(tn^{k−1}),  (26)

where X = F(G_{1}) \ F(G_{0}).

Let Z = V^{(k−1)} \ ∂G_{1}. We claim that

|Z| = O(t^{1/2}n^{k−3/2}).  (27)

Suppose not. Then the average value of z(D) = |{E ∈ Z : E ⊃ D}| over all D ∈ V^{(k−2)} is greater than O(t^{1/2}n^{1/2}). For any E, E′ ∈ Z with |E ∩E′| = k −2 we have F = E ∪ E′ ∉ X, because otherwise at least one of E, E′ ∈ ∂F is covered by an edge of S′ ∈ S(F) which then is necessarily an edge of G_{1} (as it intersects F ∈ F(G_{1}) \ F(G_{0}) in k − 1 vertices). Therefore, we have at least \binom{k}{2}^{−1} ∑_{D∈V^{(k−2)}} \binom{z(D)}{2} k-sets not in X, which exceeds \binom{n}{k−2}O(tn) by the convexity of \binom{x}{2}. This contradicts (26) and proves the claim.

Let

g_{1}(E) = |{F ∈ E(G_{1}) : F ⊃ E}|, E ∈ ∂G_{1}.

Take any F ∈ E(G_{1}). Let ∂F = {E_{1}, …, E_{k}}. We claim that all but at most two of g_{1}(E_{i})’s are larger than t/6. Suppose not, say g_{1}(E_{i}) ≤ t/6, i = 1, 2, 3. Take F′ ∈ F(F) \ F(G_{0}) and any S′ ∈ S(F′) containing F as an edge. Let F′ = E_{i} ∪ {x}, some i ∈ [k], x ∈ V \ F. The star S′ contains k − 2 edges of the form E_{j} ∪ {x}, j ≠ i. These edges cannot be in G_{0} and so contribute at least 1 to g_{1}(E_{1}) + g_{1}(E_{2}) + g_{1}(E_{3}). In total, each {x} ∪ E_{j} ∈ E(G_{1}) is counted at most twice. (Once it occurs then at most 2 edges of the form {x} ∪ E_{i} can belong to E(\overline{G}).) But this contradicts (25). The claim is proved.

Define

W = {E ∈ ∂G_{1} : g_{1}(E) ≤ m − k − 1},
T = {F ∈ E(G_{1}) : W ∩ ∂F ≠ ∅}.

We claim that |W | = O(t^{1/2}n^{k−3/2}). Suppose not. Note that for E, E′ ∈ W with |E ∩ E′| = k − 2 we necessarily have F = E ∪ E′ ∉ X for otherwise in an S′ ∈ S(F) centered at x, say x ∈ E, there are m−k edges (necessarily in E(G_{1})) different from F and covering E. Thus there are at least \binom{k}{2}^{−1} ∑_{D∈V^{(k−2)}} \binom{w(D)}{2} edges not in X, where w(D) = |{E ∈ W : E ⊃ D}|, D ∈ V^{(k−2)}. Using the convexity of the \binom{x}{2}-function as before we can argue that there more than O(tn^{k−1}) edges not in X, contradicting (26). The claim is established.

Every E ∈ W is contained in at most m − k − 1 edges of G_{1}, so |T | = O(t^{1/2}n^{k−3/2}). For every F ∈ E(G_{1})\T we have ∑_{E∈∂F} \frac{1}{g_{1}(E)} ≤ \frac{2}{m−k} +(k−2)\frac{6}{t}. Note the following easy identity

|∂G_{1}| = ∑_{F ∈E(G_{1})\T} ( ∑_{E∈∂F} \frac{1}{g_{1}(E)}) + ∑_{F ∈T} ( ∑_{E∈∂F} \frac{1}{g_{1}(E)})
≤ (\frac{2}{m − k} + O(1/t)) |G_{1}| + k|T |.

We know, see (27), that |∂G_{1}| = \binom{n}{k−1} − O(t^{1/2}n^{k−3/2}). Hence

\frac{m − k}{2} \binom{n}{k − 1} − |G| = O(t^{1/2}n^{k−3/2} + |G|/t) = O(t^{1/2}n^{k−3/2} + n^{k−1}/t).

Taking t = ⌊n^{1/3}⌋ we obtain the required result.

6.2 Sub-Designs

A t-(v, k, λ)-design (or an S_{λ}(t, k, v)) is a k-graph G of order v in which every t-set is covered by exactly λ edges. As the question whether a design exists for a given set of parameters is generally notoriously hard, one direction of research is to consider what we call here sub-designs. A t-(v, k, λ)-sub-design G is a maximal k-graph of order v such that no t-set is covered by more than λ edges. Clearly, in the latter case, G can contain at most λ\binom{v}{t}\binom{k}{t}^{−1} edges so we take e(G) as the measure of the ‘goodness’ of G.

It is easy to construct sub-designs. This can be done, for example, by starting with the empty graph and consecutively adding missing k-subsets as long as possible. If we are lucky, we obtain an S_{λ}(t, k, v); in this case e(G) is maximal possible. On the other hand, one can ask how unlucky we can be, that is, how small G can be. Let D = D(λ, t, k) be the family of all k-graphs with λ + 1 edges such that some t vertices belong to every edge. Then SAT(n, D) is the family of all sub-designs of order n. Thus we are interested in sat(n, D(λ, t, k)), the minimal size of a t-(n, k, λ)-sub-design.

Note that D(λ, 1, 2) consists of one graph, namely the star S_{λ+2}^{2} = K_{1,λ+1}. Kászonyi and Tuza [KT86] computed sat(n, S_{λ+2}^{2}). In fact, their method extends to any D(λ, 1, k).

We need the following simple lemma, whose proof we include for the sake of completeness.

Lemma 19 Given integers n′, λ and k with λ ≤ \binom{n′−1}{k−1}, there is a k-graph G′ on [n′] such that every vertex has degree λ except a set D of at most k−1 vertices of degree λ − 1.

Proof. Place the elements of [n′] clockwise on a circle to form a regular n′-gon. Define the equivalence relation ∼ on [n′]^{(k)} so that two k-sets are equivalent if some rotation maps one onto the other. (Note that we do not allow mirror reflections.) Let H_{1}, …, H_{p} ⊂ [n′]^{(k)} be the obtained equivalence classes. Clearly, each H_{i} is a regular covering of [n′] of degree which is a divisor of k. Let H_{p} be the equivalence class of the set [k] which consists of k consecutive vertices. Starting with the empty hypergraph G′ on [n′], for i ∈ [p − 1], add H_{i} to G′ if the maximal degree does not exceed λ. At the end, we will be left with some d-regular k-graph. Clearly, λ − d is at most k because otherwise we had to add every H_{i}, i ∈ [p−1], so adding H_{p} we obtain the complete k-graph on [n′], which implies the contradiction \binom{n′−1}{k−1} < λ.

Finally, we try to add some subset of H_{p} to make G′ nearly λ-regular. Take some edge E ∈ H_{p} which has not been added to G′, say E = [i + 1, i + k]. We add, one by one, the following shifts of E:

[i + 1, i + k], [i + k + 1, i + 2k], [i + 2k + 1, i + 3k], …

and so on in this order until either we come across E again or we cannot add the current edge (because then the maximal degree of G becomes larger than λ). In the former case, we take any other unused edge and repeat the procedure. In the latter case, we have the required graph built because every time the added portion of H_{p} is nearly regular, that is, the difference between the maximal and minimal degrees is always at most 1.

The following theorem gives the exact answer in almost every case, except for some small n when we have only a lower bound.

Theorem 20 Given λ ≥ 1 and k ≥ 2, let D = D(λ, 1, k) and define v by λ/k ∈ [\binom{v−1}{k−1}, \binom{v}{k−1}]. Then m-sat(n, D) ≥ \binom{v}{k} + ⌈\frac{λ(n−v)}{k}⌉. If, furthermore, λ ≤ \binom{n−v−1}{k−1} then

m-sat(n, D) = sat(n, D) = \binom{v}{k} + ⌈\frac{λ(n − v)}{k}⌉ .  (28)

Proof. Given G ∈ m-SAT(n, D), let V ⊂ V (G) consist of all vertices whose degree (that is, the number of containing it edges) is at most λ − 1. Clearly, V must span the complete k-graph, for otherwise the addition of a missing edge E ∈ V^{(k)} to G cannot create any forbidden subgraph. Thus e(G) ≥ min_{v∈[n]} f(v), where f(v) = \binom{v}{k} + \frac{λ(n−v)}{k}, which implies the lower bound on m-sat.

Conversely, let n′ = n − v and let G be the nearly λ-regular k-graph G′ on [n′] built in Lemma 19, plus the complete k-graph on V = [n′ + 1, n] and (if D = {x ∈ [n′] : d(x) < λ} ≠ ∅) plus an edge E intersecting [n′] in the set D. (Note that v ≥ k − |D| if D ≠ ∅: otherwise G′ + K^{k}(V) ∈ m-SAT(n, D) contradicts our lower bound.)

All vertices in [n′] have degree λ and any missing edge (which must intersect [n′]) creates a forbidden subgraph. Also, G is D-free: if \binom{v−1}{k−1} ≥ λ, then we obtain the contradiction f(v) ≥ λn/k > f(k − 1). The required G ∈ SAT(n, D) is built.

Next, let t = 2 and D = D(λ, 2, k), that is, we forbid λ + 1 edges having 2 common vertices. The Turán number t(n, t, k) = ex(n, K_{k}^{t}) is the maximum size of a K_{k}^{t}-free t-graph of order n. Define α(n, t, k) = \binom{n}{t} − t(n, t, k). We are able to compute asymptotically sat(n, D).

Theorem 21 Given λ ≥ 2 and k ≥ 3, let c = λ/\binom{k}{2}. Then, for any n ≥ max(k + c − 1, kc^{1/(k−2)}),

m-sat(n, D(λ, 2, k)) ≥ cα(n, 2, k).  (29)

On the other hand, for any fixed λ and k,

sat(n, D(λ, 2, k)) ≤ cα(n, 2, k) + O (\frac{n^{2}}{(log n)^{1/(\binom{k}{2}−1)}} ) .  (30)

Proof. Given a monotonically D-saturated k-graph H, we build, on the same vertex set, the 2-graph G so that {i, j} ∈ E(G) iff there are at least λ H-edges containing both i, j ∈ V (H). Clearly, any k-set E independent in G must be an edge of H, for otherwise the addition of E to H does not create a forbidden subgraph. This implies that

e(H) ≥ L(G) = k_{k}^{2}(\overline{G}) + ce(G),  (31)

where k_{k}^{2}(\overline{G}) denotes the number of K_{k}^{2}-subgraphs of \overline{G}, the complement of G. We want to find, for which 2-graphs G, the right-hand side of (31) is minimized. By a theorem of Bollobás [Bol76] (for some extensions see Schelp and Thomason [ST98]), this happens if \overline{G} is a complete multipartite 2-graph (that is, if G is a disjoint union of complete graphs). If the parts are of sizes n_{1} ≥ n_{2} ≥ … ≥ n_{l}, then we have to minimize

L(G) = k_{k}^{2}(\overline{G}) + ce(G) = ∑_{A∈[l]^{(k)}} ∏_{i∈A} n_{i} + c ∑_{i=1}^{l} \binom{n_{i}}{2},  (32)

given the condition ∑_{i=1}^{l} n_{i} = n.

Suppose that l ≥ k. Let G′ be obtained from G by merging the smallest two parts together. This adds n_{l−1}n_{l} extra edges to G, but this eliminates all K_{k}^{2}-subgraphs of \overline{G} intersecting both of the affected parts, that is,

k_{k}^{2}(\overline{G}) − k_{k}^{2}(\overline{G′}) = n_{l−1}n_{l} ∑_{A∈[l−2]^{(k−2)}} ∏_{i∈A} n_{i}.  (33)

We claim that ∑_{A∈[l−2]^{(k−2)}} ∏_{i∈A} n_{i} ≥ c. As n_{l} and n_{l−1} are two smallest parts, it is enough to verify the inequality for n_{2} = … = n_{l−2} = n_{l−1} = n_{l} = x in which case it reduces to to

g(x) = \binom{l − 3}{k − 2}x^{k−2} + \binom{l − 3}{k − 3}(n − (l − 1)x)x^{k−3} ≥ c.  (34)

Taking the derivative, one can see that the minimum of g over x ∈ [1, n/l] is achieved either for x = 1 or for x = n/l. For x = 1, the right-hand side of (34) is h(l) = \binom{l−3}{k−2} + \binom{l−3}{k−3}(n − l + 1) and, for any l ∈ [k, n], the inequality h(l) ≥ c is true as h(k) ≥ c and

h(l + 1) − h(l) = \binom{l−3}{k−3} + (n − l + 1)\binom{l−3}{k−4} − \binom{l−2}{k−3} = (n − l)\binom{l−3}{k−4} ≥ 0.

For x = n/l,

g(n/l) = \binom{l − 2}{k − 2} (\frac{n}{l})^{k−2} = \frac{n^{k−2}}{(k − 2)!} ∏_{i=1}^{k−2} (1 − \frac{i + 1}{l})

which is clearly minimized for l = k. But g(n/k) ≥ c by our assumptions.

Thus we may assume that l ≤ k−1. But then k_{k}^{2}(G) = 0 and e(G) is minimal if we have exactly k − 1 parts of nearly equal sizes (i.e. if \overline{G} is the Turán graph) and (29) follows.

To demonstrate the upper bound we have to use as bricks almost optimal sub-designs. Rödl [Röd85] was first to show that for fixed λ, k, t there exists a t-(v, k, λ)-sub-design with λ\binom{v}{t}/\binom{k}{t}+o(v^{t}) edges, v → ∞, that is, asymptotically approaching the absolute upper bound. The error term was made more specific by Gordon, Kuperberg, Patashnik and Spencer [GPKS96] who showed it to be O(F (v)), where F (v) = v^{t}/(log v)^{1/D} and D = \binom{k}{t} − 1. Gordon, Kuperberg and Patashnik [GKP95] present a few different methods suitable for practical construction of nearly optimal sub-designs.

Let us construct G ∈ SAT(n, D) showing that (29) is asymptotically correct. Partition [n] = V_{1} ∪ … ∪ V_{k−1} into k − 1 nearly equal parts. On each part V_{i} construct a maximum 2-(|V_{i}|, k, λ)-sub-design H_{i}. The union of H_{1}, …, H_{k−1} is obviously D-free and has the size within O(F (n)) of (29). Completing it in an arbitrary way to G ∈ SAT(n, D), we add O(F (v)) extra edges as each extra edge intersect some part in at least 2 vertices while each H_{i} has O(F (v)) 2-sets covered by strictly less than λ edges. The theorem is proved.

Finally, let us consider the general case t ≥ 3. It seems that sat(n, D(λ, t, k)) is generally related to α(n, t, k).

Theorem 22 For any fixed λ, t and k,

m-sat(n, D(λ, t, k)) ≥ (1 − o(1))λα(n, t, k)\binom{k}{t}^{−1},  (35)

as n tends to infinity.

Proof. Let H ∈ m-SAT(n, D(λ, t, k)). Let the t-graph G consist of all t-sets covered by at least λ edges of H. Similarly to the above, we note that any k-subset of [n] not spanning an edge in G, must belong to E(H) and therefore,

e(H) ≥ λe(G)\binom{k}{t}^{−1} + k_{k}^{t}(\overline{G}).  (36)

If e(\overline{G}) ≤ (1 + o(1))t(n, t, k) then the first summand in the right-hand side of (36) itself gives the desired lower bound. Otherwise, the result of Erdős and Simonovits [ES83] implies that the second summand is Θ(n^{k}) which is far more than required.

We do not have many structural results related to the Turán problem for complete hypergraphs. Sidorenko [Sid95] mentions the following conjectures.

α(n, 3, k) = (\frac{2}{k − 1})^{2} \binom{n}{3} + o(n^{3}),  (37)
α(n, 4, 5) = \frac{5}{16}\binom{n}{4} + o(n^{4}).  (38)

Recall that α(n, t, k) = \binom{n}{t}−t(n, t, k) is the minimum size of an α(n, t, k)-graph, that is, a t-graph on n vertices in which any k-set spans at least one edge.

Example 23 Let D = D(λ, 3, k), where either k = 4 or k ≥ 5 is odd. Then there is a D-saturated k-graph H with λ(\frac{2}{k−1})^{2}\binom{n}{3}/\binom{k}{3} + o(n^{3}) edges. In particular, if (37) is true, then H is asymptotically extremal.

Proof. Let k = 4. Let m = ⌊n/3⌋. Define A_{i} = [(i − 1)m + 1, im], i ∈ [3]. The graph G on [3m] consisting of all triples {x, y, z} with x, y ∈ A_{i} and z ∈ A_{i}∪A_{i+1}, where A_{4} = A_{1}, is an α(3m, 3, 4)-graph with approximately \frac{4}{9} \binom{n}{3} edges.

Consider the graph H′ consisting of edges E = {w, x, y, z} with {x, y, z} ∈ A_{i}^{(3)} and w ∈ A_{i+1} (then E^{(3)} ⊂ E(G)), i ∈ [3], such that u + x + y + z is congruent modulo m to an element in [λ]. Let D ∈ E(G). For example, suppose that D consists of x, y ∈ A_{1} and w ∈ A_{2}. To find z with {w, x, y, z} ∈ E(H′) we have to satisfy w + x + y + z ≡ j (mod m) for some j ∈ [λ]; there are λ solutions, but we may have to discard possible degenerate cases when z = x or z = y. A similar claim is true if D ⊂ A_{i}. Hence, each G-edge, except O(n^{2}) edges, is covered by exactly λ edges of H′.

It is therefore clear that if we complete the D-free graph H′ to any Dsaturated graph H on [n], then we add only O(n^{2}) edges; therefore, H has the required size.

For k = 2l + 1, l ≥ 2, an example of an α(n, 3, k)-graph G attaining (37) is obtained by partitioning [n] = A_{1} ∪ … ∪ A_{l} into nearly equal parts and letting G = ∪_{i∈[l]}K^{3}(A_{i}). The result of Rödl [Röd85] implies that we can find a D-free k-graph on each A_{i} which is a nearly optimal 3-(|A_{i}|, k, λ)-sub-design; let H′ be the union of these. Completing it arbitrarily to a D(λ, 3, k)-saturated graph, we add only o(n^{3})-extra edges, which proves the claim.

However, we do not know any matching construction for t = 3 and even k ≥ 6. In this case, a conjectured extremal α(n, t, k)-graph G is the disjoint union of complete 3-graphs plus at least one α(m, 3, 4)-extremal graph. The last graph causes us the problem: the constructions by Kostochka [Kos82] do not admit an almost perfect covering by k-edges, k ≥ 6.

Here is a short explanation why. In all Kostochka’s graphs we have three equisized sets A_{1} ∪ A_{2} ∪ A_{3} = [n] and let G = (∪_{i=1}^{3}K^{3}(A_{i})) ∪ G′, where G′ consists of \frac{1}{3} \binom{n}{3} + o(n^{3}) other edges. Also, any k-set E with E^{(3)} ⊂ E(G) has the property that |K ∩ A_{i}| ≥ k − 2 for some i, so it can cover at most l = \binom{k}{3} − \binom{k−2}{3} edges of G′. Hence, we need at least e(G′)/l covering edges, which exceeds (\frac{2}{k−1})^{2}\binom{n}{3}/\binom{k}{3} + o(n^{3}) for k ≥ 8. (For k = 6 we need a slightly more refined argument.)

For similar reasons, there is no almost perfect covering of the construction by de Caen, Kreher and Wiseman [dCKW88] which gives the upper bound in (38).

Unfortunately, we do not know any other, essentially different, constructions attaining (37) or (38) and we do not have any likely guess what sat(n, D) could be then.

6.3 Triangular Families

The notion of a triangle-free 2-graph can be extended to hypergraphs in the following way: a k-graph is triangle-free if the symmetric difference of any two distinct edges is not contained in a third edge. Clearly, this is the same as forbidding the triangular family T_{k} which consists of all k-graphs with three edges E_{1}, E_{2}, E_{3} such that E_{1}△E_{2} ⊂ E_{3}.

Katona [Kat74] raised the problem of computing ex(n, T_{3}) which was solved by Bollobás [Bol74] who showed that the complete 3-partite 3-graph with parts of nearly equal sizes is a maximum triangle-free 3-graph. Bollobás [Bol74] stated the general conjecture that the analogous construction gives ex(n, T_{k}) for any k ≥ 4; Sidorenko [Sid87] proved that this is the case for k = 4.

Concerning the sat-function, we have the following obvious example of a T_{k}- saturated graph: the pyramid P = P (k − 1, n − k + 1; k − 1, 1) which consists of all k-subsets of [n] containing the set [k − 1] called basic. Indeed, any missing edge E intersects [k, n] in at least 2 points and creates a forbidden subgraph on the set E ∪ [k − 1]. Thus

sat(n, T_{k}) ≤ n − k + 1, n ≥ k + 1

and this might be sharp. It is remarkable that P can be viewed as the complete k-partite k-graph with k − 1 parts consisting of only one vertex.

In the general case we are able to prove only the following.

Theorem 24 Let k ≥ 3 be fixed. Then

n − O(log n) ≤ sat(n, T_{k}) ≤ n − k + 1.

Proof. We have to prove the lower bound. Let G be a minimum T_{k}-saturated graph on [n]; e(G) ≤ n − k + 1. Consecutively choose G_{1}, G_{2}, … ⊂ G as follows: let e_{j+1} be the largest integer such that the k-graph H_{j}, E(H_{j}) = E(G) \ (E(G_{1}) ∪ … ∪ E(G_{j})), contains a P (k − 1, e_{j+1}; k − 1, 1)-subgraph and let G_{j+1} be any such subgraph. We terminate the procedure when b_{j} = n−e_{[j]} −j(k −1) is less than max(j, k). (We denote e_{[j]} = ∑_{i∈[j]} e_{i}, etc.)

Let j ≥ 0 and suppose we have chosen G_{1}, …, G_{j}. Let B_{j} consist of some b_{j} vertices not covered by an edge of G_{i}, i ∈ [j]; B_{j} exists as v(G_{i}) = e_{i} + k − 1. (We let b_{0} = n.) Label all (k − 1)-subsets of [n] by A_{1}, …, A_{l}, l = \binom{n}{k−1}. Let d_{i} be the number of edges of H_{j} containing A_{i}, i ∈ [l]. Clearly,

d_{[l]} = ke(H_{j}) ≤ k(n − k + 1 − e_{[j]}) = k(b_{j} + (j − 1)(k − 1)) < k^{2}b_{j}. (39)

The number of ways to add an element of B_{j}^{(k)} creating a forbidden subgraph with any given E_{1}, E_{2} ∈ [n]^{(k)} is at most \binom{b_{j}−2}{k−2} + O(1) if |E_{1} ∩ E_{2}| = k − 1 and it is O(b_{j}^{k−4}) otherwise. As the addition of any E ∈ B_{j}^{(k)} \ E(H_{j}) to H_{j} creates a forbidden subgraph (because E is disjoint from any edge of G_{i}, i ∈ [j]), we conclude that

O(b_{j}^{k−4})\binom{e(H_{j})}{2} + \binom{b_{j} − 2}{k − 2} ∑_{i∈[l]} \binom{d_{i}}{2} ≥ \binom{b_{j}}{k} − e(H_{j}),  (40)

and, by (39),

∑_{i∈[l]} \binom{d_{i}}{2} ≥ \frac{b_{j}^{2}}{k(k − 1)} − O(b_{j}).  (41)

We have e_{j+1} = max_{i∈[l]} d_{i}. The convexity of the \binom{x}{2}-function implies that the left-hand side of (41) does not exceed \frac{d_{[l]}}{e_{j+1}} \binom{e_{j+1}}{2} < \frac{1}{2}k^{2}b_{j}e_{j+1}. Therefore, we obtain that

e_{j+1} ≥ \frac{2b_{j}}{k^{3}(k − 1)} − O(1).

From this inequality (and from the fact that e_{j+1} ≥ 1 if b_{j} ≥ k) we deduce the following inequality

b_{j+1} ≤ min ((1 − \frac{2}{k^{3}(k−1)}) b_{j} + O(1), b_{j} − k) .  (42)

It is clear that, starting with b_{0} = n, we stop after j = O(log n) steps. Now,

e(G) ≥ e_{[j]} = n − b_{j} − j(k − 1) = n − O(log n).

The theorem is proved.

Let us consider the case k = 3; note that T_{3} contains only 2 non-isomorphic graphs, S_{4}^{3} and T_{3}:

E(S_{4}^{3}) = { {1, 2, 3}, {1, 2, 4}, {1, 3, 4} },
E(T_{3}) = { {1, 2, 3}, {1, 2, 4}, {3, 4, 5} }.

Theorem 25 For any n ≥ 4, sat(n, T_{3}) = n − 2.

Proof. Let G be any T_{3}-saturated graph on [n]. Make a list of all edges of G and, consecutively and as long as possible, merge together any two sets in the list sharing at least 2 vertices (that is, replace then by their union.) Call the resulting sets C_{1}, …, C_{l} ⊂ [n] components. Let v_{i} = |C_{i}|. Define the 2-graph H on [n] by

E(H) = {{x, y} ∈ [n]^{(2)} : {x, y} = E_{1}△E_{2} for some E_{1}, E_{2} ∈ E(G)}.

Consider any component C. It is easy to see by induction on |C| that C is composed of at least |C| − 2 edges of G.

Note that if E ∈ E(H[C]) then any E_{1}, E_{2} ∈ E(G) with E_{1}△E_{2} = E share 2 vertices and so belong to the same component C′; but E ⊂ C′ ∩C so necessarily C′ = C. Claim 1 For every component C, Δ(H[C]) ≤ e(G[C]) − 1.

Let x ∈ C be arbitrary. For each {x, y} ∈ E(H[C]), choose D_{y}, E_{y} ∈ E(G) with D_{y}△E_{y} = {x, y} and E_{y} ∋ y. If {x, z} is another edge of H[C] then E_{y} ≠ E_{z}: indeed, otherwise D_{z}△E_{z} = {x, z} ⊂ D_{y} and G contains a forbidden subgraph. Hence, d(x) ≤ e(G[C]) − 1 (we must have at one G-edge incident to

x) and the claim is proved. Claim 2 If e(G[C]) ≤ |C| − 1 then for any x ∈ [n] \ C there is a component C′ ∋ x intersecting C.

By Claim 1, there exists \{a, b\} ∈ E(\overline{H}[C]). As x ∉ C, E = \{a, b, x\} ∉ E(G). Consider a forbidden subgraph F created by E. We are home if {a, x} or {b, x} is covered by E_{1} or E_{2}, where E(F) = {E, E_{1}, E_{2}}. If {a, b, y} ∈ E(F) then y ∈ C and the remaining edge of F contains both x and y. Finally, if E_{1}△E_{2} ⊂ E then, as {a, b} ∉ E(H), x belongs to the component containing E_{1} and E_{2} which is the required one.

The claim is proved. In particular, C_{[l]} = V (G).

Now, if every component C spans at least |C| − 1 edges then we are home: by Claim 2 relabel components C_{1}, …, C_{l} so that C_{i} ∩ C_{[i−1]} ≠ ∅, i ∈ [2, l], and it is easy to show by induction on i that C_{[i]} is made of at least |C_{[i]}| − 1 edges, which gives e(G) ≥ n − 1.

So, suppose that, for example, e(G[C_{1}]) = |C_{1}|−2. If for every x ∈ V (G)\C_{1}, there are two distinct components containing x and intersecting C_{1} then are home:

e(G) ≥ ∑_{i∈[l]}(v_{i} − 2) = v_{1} − l − 1 + ∑_{i∈[2,l]}(v_{i} − 1)
≥ v_{1} − l − 1 + max(2l − 2, 2(n − v_{1})) ≥ n − 2.  (43)

So let C_{2} be the only component containing some vertex x ∉ C_{1} and intersecting C_{1}. Let {y} = C_{1} ∩ C_{2}. Let z ∈ V (G) \ C_{[2]} be arbitrary. (The below argument works without any changes if C_{1} ∪ C_{2} = V (G).)

If {x, z} ⊂ C_{i}, for some i ∈ [3, l], then, by the choice of x, C_{i} ∩ C_{1} = ∅ and, by Claim 2, there exists another component through z intersecting C_{1}.

If no component contains both x and z then, for every y′ ∈ C_{1} \ {y}, E = {x, y′, z} ∉ E(G) and considering a forbidden subgraph created by E we conclude that, for some i ∈ [3, l], {y′, z} ⊂ C_{i} (as {x, y′} cannot lie within a component by the definition of x). As |C_{1}| ≥ 3, we have at least 2 distinct components containing z and intersecting C_{1}.

Now the argument similar to (43) shows that C_{[3,l]} is made of at least n − |C_{1} ∪ C_{2}| edges, which gives e(G) ≥ n − 3.

Can we have e(G) = n − 3? If we have the equality then every C_{i}, i ∈ [3, l], must intersect C_{1} ∪ C_{2} in exactly one vertex and e(G[C_{j}]) = |C_{j}| − 2, j ∈ [l]. By Claim 1, there exists y_{i} ∈ C_{i} such that {y, y_{i}} ∉ E(H), i = 1, 2. But then {y, y_{1}, y_{2}} ∉ E(G) (e.g. because it intersects C_{1} in two vertices) and the consideration of a created forbidden graph yields a component containing both y_{1} and y_{2}. Hence, e(G) > n − 3 as required.

Remark. Our further analysis has not yet yielded any characterization of the cases of equality: we have got stuck considering different cases and, even if we had succeeded, the proof would have been rather long. Therefore, we present only some other constructions which we have discovered in our search. First, there is another minimum T_{3}-saturated graph of order 7: let V (G) = [7] and

E(G) = { {1, 2, 5}, {1, 3, 6}, {1, 4, 7}, {2, 3, 4}, {5, 6, 7} }.

Also, concerning the m-sat-function, we have yet another construction with n−2 edges for any n ≥ 6: add, to the pyramid P (2, n − 4; 2, 1) with basic vertices a, b, new vertices x, y and new edges {x, y, a} and {x, y, b}.

6.4 K_{m}-Saturated Graphs

Duffus and Hanson [DH86] consider sat(n, K_{m}, l) which is the minimum size of

G ∈ SAT(n, K_{m}, l) = {G ∈ SAT(n, K_{m}) : δ(G) ≥ l}.

Of course, any K_{m}-saturated graph G has minimal degree at least m − 2, so we assume l ≥ m − 1.

Duffus and Hanson [DH86] proved that, for n ≥ 5, sat(n, K_{3}, 2) = 2n−5 and, for n ≥ 10, sat(n, K_{3}, 3) = 3n − 15. However, their general lower bound [DH86, Theorem 2], which states that sat(n, K_{m}, l) ≥ \frac{l+m−2}{2}n + O(1), is far from the actual value. Trying to improve this bound, we showed that sat(n, K_{m}, l) = ln+ O(\frac{n log log n}{log n}) for any fixed l ≥ m−1. Later, we learned that Alon, Erdős, Holzman and Krivelevich [AEHK96, Theorem 2] showed that any G ∈ SAT(n, K_{m}) with O(n) edges has an independent set of size n − O(\frac{n}{log log n}), which implies that sat(n, K_{m}, l) = ln + O(\frac{n}{log log n}). However, we decided to present our proof because it improves all these bounds and we think that our general Theorem 26 is of independent interest.

However, the question of Bollobás [Bol95, p. 1271] whether sat(n, K_{3}, l) = ln + O(1) for any fixed l ≥ 4, remains open.

Let us give a construction of G ∈ SAT(n, K_{m}, l) with ln + O(1) edges: take G = K_{m−3} + K_{l−m+3,n−l} which has minimal degree l for n ≥ 2l − m + 3. The complete bipartite graph K_{l−m+3,n−l} does not contain a triangle but the addition of any new edge violates this; hence, G is K_{m}-saturated.

To prove our lower bound we need some preliminaries. Given any d, define a_{d−m+2} = 2 and, consecutively for j = d − m + 1, d − m, …, 1, 0,

c_{j+1} = (m − 2)(a_{j+1} − 1) + 1
b_{j+1} = (m − 2)(c_{j+1} − 1) + 1
b′_{j+1} = \binom{d−j−1}{m−2}(b_{j+1} − 1) + 1,
a_{j} = \binom{d−j−1}{m−2}(b′_{j+1} − 1) + 2.

Finally, let a = (1 + 2(d − 1) + 2(d − 1)^{2})a_{0}.

Given a K_{m}-saturated graph G, let A denote the set of G-edges connecting two vertices of degree at most d in G:

A = {{x, y} ∈ E(G) : d(x) ≤ d, d(y) ≤ d}.

The following theorem states that the size of A is bounded by a = a(d, m) which does not depend on n. Note that we do not impose any restriction on the minimal degree of G.

Theorem 26 For any G ∈ SAT(n, K_{m}), m ≥ 3, we have |A| < a.

Proof. Suppose, on the contrary, that |A| ≥ a.

We prove, by induction on j = 0, 1, …, d − m + 2, that we can find the following configuration in G: a_{j}-sets X_{j} and Y_{j} and j-sets U_{j} and V_{j} (all disjoint) such that (i) X_{j} ∪Y_{j} induces in G exactly a_{j} edges which form a perfect matching between X and Y and belong to A; (ii) Γ_{U_{j}∪V_{j}}(x) = U_{j} for any x ∈ X_{j} and Γ_{U_{j}∪V_{j}}(y) = V_{j} for any y ∈ Y_{j}.

For j = 0 (when U_{0} and V_{0} are empty), we take, one by one, edges from A. Once we have selected an edge E ∈ A, cross out all incident to E edges (at most 2(d − 1) edges) and their neighbouring edges (of which at most 2(d − 1)^{2} can belong to A). Hence, we can build an induced matching of size at least |A|/(1 + 2(d − 1) + 2(d − 1)^{2}) ≥ a_{0} as required.

Suppose that j ∈ [0, d − m + 1] and we have X_{j}, etc., constructed. Choose x ∈ X_{j}; it has already got j +1 neighbours in G: the neighbour y ∈ Y_{j} plus all j vertices of U_{j}. Let N_{x} denote the remaining neighbours of x; thus |N_{x}| ≤ d−j−1. For any z ∈ Y_{j} distinct from y, the addition of the edge {x, z} must create a copy of K_{m}, say on a set D_{z} ∪ {x, z}. Now, D_{z} ⊂ Γ(x) ∩ Γ(z) ⊂ N_{x}.

Thus some set D_{z}, z ∈ Y_{j} \ \{y\}, appears at least b′_{j+1} = ⌈(a_{j} − 1)/\binom{d−j−1}{m−2}⌉ times; suppose it is D ∈ N_{x}^{(m−2)} which equals D_{z} for z ∈ B′ ⊂ Y_{j} \ {y}, |B′| = b′_{j+1}. In a similar manner, we try to connect y to the X_{j}-matches of B′-vertices and find a set E ∈ N_{y}^{(m−2)} spanning the complete graph and connected to every z from a set B ⊂ X_{j} matched into B′ of cardinality b_{j+1} = ⌈b′_{j+1}/\binom{d−j−1}{m−2}⌉.

Clearly, no z ∈ B can be connected to every vertex of D; otherwise D, z and the match of z in B′ span K_{m}. Therefore, some v ∈ D is not connected to at least c_{j+1} = ⌈\frac{b_{j+1}}{m−2}⌉ vertices of B; let C ⊂ B consist of all such vertices. Similarly, we can find u ∈ E, not connected to an a_{j+1}-set Y_{j+1} matched into C. Of course, u ≠ v. Now, let U_{j+1} = U_{j} ∪{u}, V_{j+1} = V_{j} ∪{v}, and let X_{j+1} ⊂ X_{j} consist of the matches of Y_{j+1}, which completes our induction.

At the end, we try to apply our argument again, for j = d − m + 2. We obtain that x ∈ X_{j} has at least 1+j +(m−2) > d neighbours, which contradicts the fact that {x, y} ∈ A, where y is the Y_{j}-match of x.

Now we are ready to improve the result of Alon et al [AEHK96, Theorem 2] mentioned above. Let α(G) denote the maximal size of independent Y ⊂ V (G).

Lemma 27 For any G ∈ SAT(n, K_{m}) with O(n) edges, we have

α(G) = n − O (\frac{n log log n}{log n}) .

Proof. Suppose e(G) ≤ Cn. Let d = \frac{ε log n}{log log n} for some fixed ε > 0 and let X = {x ∈ V (G) : d(x) > d}. Now, d|X|/2 ≤ e(G) ≤ Cn implies that

|X| ≤ \frac{2Cn log log n}{ε log n}.

By Theorem 26, Y = V (G)\X spans at most a ≤ n^{2ε(m−2)+o(1)} edges. Removing at most a vertices we can make Y independent; it has the required size if ε < \frac{1}{2(m−2)}.

Clearly, e(G) ≥ α(G)δ(G). Therefore, Lemma 27 implies the following result.

Theorem 28 For any fixed l ≥ m − 1, sat(n, K_{m}, l) = ln + O(\frac{n log log n}{log n}).

Part II

Weakly Saturated Hypergraphs

7 Introduction

In this part we move to studying weakly-saturated graphs. They are briefly mentioned in Section 1 which also contains an example how such a notion can naturally appear in real-life problems.

Let us give some basic definitions, describe what is known about the w-sat-function, and indicate which new results are obtained in this part.

7.1 Definitions

Let F be a family of forbidden r-graphs. An r-graph G of order n is called weakly F-saturated, denoted G ∈ w-SAT(n, F), if we can consecutively add all missing edges to G so that each time we add an edge at least one new forbidden subgraph appears. Such an ordering of E(\overline{G}) is called F-proper. Equivalently, G ∈ w-SAT(n, F) if the weak closure Cl_{F}(G) is the complete r-graph on V (G). (The weak closure is obtained by taking the iterated (strong) F-closure (defined in Subsection 4.4) until it stabilizes: Cl_{F}(G) = Cl_{F}^{∗}(...(Cl_{F}^{∗}(G))...).) We are generally interested in

w-sat(n, F) = min{e(G) : G ∈ w-SAT(n, F)}.

Note that we do not require that G is F-admissible as this does not affect w-sat(n, F): if G ∈ w-SAT(n, F) contains a forbidden subgraph F ⊂ G, then the graph obtained from G by the removal of any F -edge is still weakly F-saturated, so G cannot be minimal. Clearly, w-sat(n, F) ≤ m-sat(n, F) ≤ sat(n, F).

If the forbidden family consists of only one member, F = {F }, then we use the shortcuts w-SAT(n, F) = w-SAT(n, F), etc.

7.2 Survey

Let us give a short survey of w-sat-type results. Unfortunately, not much is known about the w-sat-function.

Tuza [Tuz92] showed that, for any fixed r-graph F,

w-sat(n, F) = Θ(n^{s(F)}).  (44)

Here s(F) ∈ [0, r − 1] is what he calls the local sparseness of F:

s(F) = min{s(E) : E ∈ E(F)},  (45)

where the sparseness of an edge E ∈ E(F) is the smallest natural number s for which there is an A ⊂ E with |A| = s + 1 such that A ⊂ E′ ∈ E(F) implies E′ = E.

Alon [Alo85] proved that, for any fixed 2-graph F, the ratio w-sat(n, F)/n tends to a limit as n → ∞.

Apparently, w-sat-type problems were first considered by Bollobás [Bol67c] who made a conjecture about the value of w-sat(n, K_{m}^{2}). This conjecture was proved by Frankl [Fra82] and by Kalai [Kal84, Kal85]; the result is implicit in Lovász [Lov77]; see also Alon [Alo85]. They proved that

w-sat(n, K_{m}^{r}) = \binom{n}{r} − \binom{n − m + k}{r}, n ≥ m > k.  (46)

In fact, Alon [Alo85] proved a more general result: he computed the w-sat-function for K_{m_{1}}^{r_{1}} ⊗ … ⊗ K_{m_{t}}^{r_{t}}, where ⊗ denotes the join operator defined in Section 12. (A different proof of Alon’s result is presented by Yu [Yu93].)

Kalai [Kal85] showed that, for the complete bipartite graph K_{st},

w-sat(n, K_{st}) ≥ (s − 1)n − \binom{s − 1}{2}, 2 ≤ s ≤ t,  (47)

which is sharp for s = t and n ≥ 3s − 2.

Kalai [Kal85] also proved that, for the wheel W_{m} = v + C_{m}, we have

w-sat(n, W_{m}) ≥ 2n − 3,  (48)

while it is easy to show that sat(n, W_{m}) ≤ 2n−3+ε, where ε = 0 if m ≤ n−2 is odd and ε = 1 if m is even or if m = n − 1, cf. Theorems 49–51 and Lemma 59.

Tuza [Tuz88, Conjecture 7] conjectured that

w-sat (n, H_{r}(r + 1, l)) = \binom{n − r + l − 2}{l − 2} n ≥ r + 1 ≥ l ≥ 3,  (49)

where the uniform family H_{r}(m, l) consists of all r-graphs of order m and size l. Clearly, H_{r}(r + 1, r + 1) = {K_{r+1}^{r}}, so (46) implies (49) for l = r + 1. The case l = 3 of Tuza’s conjecture was settled by Erdős, Füredi and Tuza [EFT91].

These were perhaps all known results on w-sat(n, F) for non-trivial specific instances of F.

7.3 Our Approach

The characteristic feature of w-sat-type problems is that, given a particular forbidden family F, it is usually fairly easy to come up with a correct example of G_{n} ∈ w-SAT(n, F), which gives us an upper bound on w-sat(n, F). (And, as a rule, we have many different extremal graphs.) However, it is usually very hard to prove the matching lower bound. So, techniques for establishing lower bounds are of importance.

The notion of dependence turned out to be useful; for example, all proofs of (46) exploit some form of it. This approach was most clearly formulated by Kalai [Kal85]: if we have a matroid M on [n]^{(r)} such that any F ∈ F is an M-chain, then

w-sat(n, F) ≥ R_{M}([n]^{(r)}),  (50)

the rank of M. (An r-graph F is an M-chain if, for any embedding V (F) ⊂ [n], any edge E ∈ E(F) lies in the M-span of E(F) \ {E}.) See Lemma 33 for a proof of (50).

We base our approach (which is described in detail in Section 9) on this idea; we exploit what we call gross and count matroids.

Gross matroids are constructed by means of exterior algebra. They were considered by Kalai [Kal90] (but for other purposes); we define them in Subsection 8.1. In brief, the gross matroid G_{G} of an r-graph G is a matroid on r-uniform set systems with G being a base; thus its rank is e(G). Now, if every F ∈ F is a G_{G}-chain, then

w-sat(n, F) ≥ R_{G_{G}}([n]^{(r)}).  (51)

The lower bound (51) is said to be g-proved. If this method gives the actual value of w-sat(n, F), then we say that F admits a g-proof for n. A related method (g′-proof) is also introduced.

The principal difficulty of the matroid approach (50) is that it is not clear at all how to search for a suitable matroid M. However, if we have G ∈ w-SAT(n, F) conjectured to be minimal, then G_{G} is a good candidate for M. If each forbidden graph is a G_{G}-chain, then, by (51), we know w-sat(n, F) exactly. In this case we say that the pair (F, G) admits a G-proof.

Our count matroid is a general and natural extension of the construction by White and Whiteley [WW84], see Subsection 8.2. For example, our count matroids admit many polynomials in n as the rank function while the original definition yielded linear functions only. If M in (50) is a count matroid, then the lower bound (50) is said to be c-proved. If the bound is sharp, then F admits a c-proof for n. Here as well, if we have a conjecture on w-sat(n, F), then there is one particular count matroid which is worth looking at; if this method works, then we have a C-proof.

Unfortunately, our approaches do not always succeed: we can indicate many concrete pairs (F, G) not admitting a C/G-proof with G ∈ w-SAT(n, F) being minimal. However, using these techniques we have managed to prove many new results which we are going to describe now.

Given sequences of integers s = (s_{1}, …, s_{t}) and r = (r_{1}, …, r_{t}), the pyramid P (s; r) is the r-graph, r = r_{1} + … + r_{t}, with vertex set being the disjoint union S_{1} ∪…∪S_{t}, |S_{i}| = s_{i}, and with the edge set consisting of those r-subsets which, for every i ∈ [t], intersect S_{1} ∪ … ∪ S_{i} in at least r_{1} + … + r_{i} vertices. The notion of a pyramid is rather general: we obtain, as partial cases,

K_{m}^{r} = P (m; r),
S_{m}^{r} = P (1, m − 1; 1, r),
K_{l} + E_{m} = P (l, m; 1, 1),
H_{r}(r + 1, l) = P (r − l + 1, l; r − l + 1, l − 1),

and more. Instances of pyramids appear explicitly quite often in the literature.

Applying gross matroids, we compute w-sat(n, P (s; r)) for all feasible sets of parameters n, s and r, see Subsection 10.1. Among other things, this implies (46) and computes w-sat(n, H_{r}(r + 1, l)), confirming the formula (49) conjectured by Tuza [Tuz88, Conjecture 7].

Erdős, Füredi and Tuza [EFT91] asked for a description of all minimum weakly H_{r}(r + 1, 3)-saturated graphs. In general, G/g/g′-proofs do not provide any good characterization of the cases of equality, but our Theorem 44 does this for H_{r}(r + 1, 3) by providing a different (combinatorial) proof which employs some ideas from [EFT91]. (In fact, H_{r}(r + 1, 3) admits a C-proof.) In Section 10.2 we provide a construction of G ∈ w-SAT(n, H_{r}(m, l)), for all n, k, l and m, which we conjecture to be minimal. Applying count matroids, we determine more values of w-sat(n, H_{r}(m, l)). Applying the g′-proof technique, we compute exactly w-sat(n, H_{2}(m, l)) for all possible n, m and l and obtain some asymptotic results. Also, we observe that we have incidentally computed (with a g′-proof) the w-sat-function for any initial segment of [n]^{(2)} in the colex order.

Our more general results of Section 11 imply in particular that if (F, G) admits a G/g/g′-proof and every r − 1 vertices of F are covered by an edge, then the pair (cn(F), cn(G)) admits a G/g/g′-proof. (The cone cn(F) of an kgraph F is obtained by adding to F a new vertex v and all r-edges containing v.)

In the class of 2-graphs, for example, we have cn^{l}(F) = K_{l}+F. The following 2-graphs are shown to admit a G/g/g′-proof: complete graphs, stars, odd cycles, initial colex-segments of [n]^{(2)}, disjoint edges, paths (more generally, almost every forest or tree), and some others; please refer to Subsection 10.3 for details. Therefore, we are able to compute the w-sat-function for K_{l} + F, where F is any of these graphs.

Note that cn(K_{m}^{r}) = K_{m+1}^{r} and K_{r}^{r}, the single edge, trivially admits a Gproof as w-sat(n, K_{r}^{r}) = 0. This shows that complete graphs admit a G-proof and gives another proof of (46).

In Section 12 we define the ⊗-operator, which we call join, and prove among other things that if every pair (F_{i}, G_{i}), i ∈ [t], admits a G/g-proof, then so does the pair (F_{1} ⊗ … ⊗ F_{t}, G), where \overline{G} = \overline{G_{1}} ⊗ … ⊗ \overline{G_{t}}. As complete graphs admit a G-proof, the computation of the w-sat-function for joins of complete graphs by Alon [Alo85] (another proof is presented by Yu [Yu93]) is a special instance of our result. By applying the join operator, we can indicate many new graphs for which we can compute the w-sat-function exactly.

8 Matroids

Here we define gross and count matroids and establish their basic properties. (For an introduction to matroid theory, we refer the reader to the texts by Welsh [Wel76] or Oxley [Oxl92].) Our approach to w-sat-type problems, which exploits these notions, is described in Section 9.

8.1 Gross Matroids

Here we define the notion of a gross matroid by means of exterior algebra. Some background in multilinear algebra is included; for a more comprehensive treatment of the topic, the reader may consult Bourbaki [Bou74] or Marcus [Mar75].

8.1.1 Exterior Algebra

Let V be an n-dimensional real vector space with a basis e = {e_{1}, …, e_{n}}. Its exterior algebra ⋀V is the 2^{n}-dimensional vector space with the formal basis (e_{A})_{A⊂[n]}. (We identify e_{i} with e_{{i}} and e_{∅} with the scalar 1 ∈ ℝ.) It comes equipped with an associative bilinear ∧-product which is completely determined by

\begin{matrix}e_{i} ∧ e_{j} = −e_{j} ∧ e_{i}, & i, j ∈ [n], \\ e_{v_{1}} ∧ … ∧ e_{v_{k}} = e_{\{v_{1},...,v_{k}\}}, & 1 ≤ v_{1} < … < v_{k} ≤ n. \end{matrix}

Let (e_{A}^{∗})_{A⊂[n]} be the dual basis of (e_{A})_{A⊂[n]}. We naturally identify ⋀(V^{∗}) and (⋀V)^{∗} so that e_{v_{1}}^{∗} ∧ … ∧ e_{v_{k}}^{∗} corresponds to e_{{v_{1},...,v_{k}}}^{∗}, 1 ≤ v_{1} < … < v_{k} ≤ n.

Let f = (f_{1}, …, f_{n}) be another basis of V; in the obvious way we define f_{A}, f_{A}^{∗} for A ⊂ [n], etc. By M = (α_{ij})_{i,j∈[n]} we denote the n × n-matrix satisfying f^{∗} = Me^{∗}, that is,

f_{i}^{∗} = α_{i1}e_{1}^{∗} + … + α_{in}e_{n}^{∗}, i ∈ [n].

Assume that f is in the generic position with respect to e, that is, the entries of M are n^{2} transcendentals algebraically independent over the rationals. An alternative definition is to assume that the entries are n^{2} independent variables; any equation we will consider can be reduced to the form P = 0 for some polynomial P in the α’s with integer coefficients and we agree that the statement is true if and only if P is the zero polynomial.

Let ⋀^{i}V be the subspace of ⋀V spanned by (e_{A})_{A∈[n]^{(i)}}. We denote

⟨g^{∗}, h⟩ = g^{∗}(h), g^{∗} ∈ ⋀V^{∗}, h ∈ ⋀V.

For g^{∗} ∈ ⋀V^{∗}, h ∈ ⋀V, the left interior product g^{∗} ⌞ h ∈ ⋀V is defined by

⟨u^{∗}, g^{∗} ⌞ h⟩ = ⟨u^{∗} ∧ g^{∗}, h⟩, for all u^{∗} ∈ ⋀V^{∗}.

Thus, if g^{∗} ∈ ⋀^{d}V^{∗} and h ∈ ⋀^{d+l}V then g^{∗} ⌞ h ∈ ⋀^{l}V, d, l ≥ 0. One can easily check that ⌞ is a bilinear function, such that u^{∗} ⌞ (g^{∗} ⌞ h) = (u^{∗} ∧ g^{∗})⌞ h and, for the basis vectors, we have:

e_{A}^{∗} ⌞ e_{B} = \{ \begin{matrix}±e_{B\A}, & if A ⊂ B, \\ 0, & if A ⊄ B. \end{matrix}

(The actual signs of ±1-coefficients do not interest us at all.) Note that by the generality of f we have ⟨f_{F}^{∗} , e_{E}⟩ ≠ 0 for any E, F ∈ [n]^{(r)}. Moreover, for any |E| = r, f ∈ ⋀^{i}V and g ∈ ⋀^{r−i}V, we have

⟨e_{E}^{∗}, f ∧ g⟩ = ∑_{A∈E^{(i)}} σ_{A,E}⟨e_{A}^{∗}, f⟩ · ⟨e_{E−A}^{∗}, g⟩,  (52)

where σ_{A,E} = ±1 depending on A and E.

For h ∈ ⋀V, its support is defined by

supp(h) = {A ⊂ [n] : e_{A}^{∗}(h) ≠ 0}.  (53)

That is, to find supp(h), write h = ∑_{A⊂[n]} c_{A}e_{A} and take those A ⊂ [n] for which the corresponding coefficient is non-zero. If we take the support in the basis f we emphasize this by adding a subscript:

supp_{f} (h) = {A ⊂ [n] : f_{A}^{∗}(h) ≠ 0};

while the supp alone always means the support relative to e as defined by (53).

Note that the cancellation (g^{∗}∧e_{A}^{∗})⌞ (h∧e_{A}) = g^{∗} ⌞ h (which is not generally correct) can be applied if, for example, each B ∈ supp(h) is disjoint from A. We will use identities like this a few times without detailed explanations. (The best way to verify them is to check them for the basis vectors.)

8.1.2 Definitions

Let us describe how to construct the gross matroid G_{G} of an r-graph G of order n.

Identify the vertices of G with the basis e = {e_{1}, …, e_{n}} of V. Let Z ⊂ ⋀^{r}V be defined by the following linear relations:

Z = {h ∈ ⋀^{r}V : f_{E}^{∗} ⌞ h = 0 for all E ∈ E(G)}.  (54)

As (f_{E}^{∗})_{E∈[n]^{(r)}} forms a basis for ⋀^{r}V^{∗}, we conclude that the e(G) relations defining Z are linearly independent so dim Z = \binom{n}{r} − e(G) and, in fact, Z is spanned by \{f_{E} : E ∈ E(\overline{G})\}.

We define the gross matroid G_{G} on [n]^{(r)} so that an r-graph F on [n] is dependent if, for some coefficients c_{E} (not all zero), we have ∑_{E∈E(F)} c_{E}e_{E} ∈ Z. To verify this condition we have to find a non-zero solution (c_{E})_{E∈E(F)} of the following system of e(G) linear equations:

∑_{E∈E(F)} c_{E}f_{D}^{∗} ⌞ e_{E} = 0,  D ∈ E(G).  (55)

By M(G, F) we denote the e(G) × e(F)-matrix corresponding to (55). The columns of M(G, [n]^{(r)}) provide a representation of G_{G}. Note that the matroid G_{G} does not depend on the choice of generic f. Also, G_{G} is a symmetric matroid, that is, for any permutation σ : [n] → [n], A ⊂ [n]^{(r)} is G_{G}-independent if and only if σ′(A) is, where σ′ is the induced action on [n]^{(r)}. Therefore, we can apply the notion of G_{G}-dependence to an r-graph F with any vertex set. (If v(F) > v(G), we add isolated vertices to G.)

This construction is not new; Kalai [Kal90] used it to construct symmetric matroids with a given growth polynomial. Also, in the partial case G = P_{k,n−k}, the matroid G_{G} is exactly Kalai’s [Kal85] k-hyperconnectivity matroid on [n]^{(2)} which was used to compute the w-sat-function for complete graphs. These two papers by Kalai were the starting points of our research on gross matroids.

Clearly, the rank of G_{G} is codim(Z) = e(G). It is easy to show that G is a base of G_{G}. Indeed, the determinant of M(G, G) is a polynomial in the α’s which assumes value 1 when M (and then M(G, G)) is the identity matrix. Therefore, the determinant is non-zero for a generic M and the columns of M(G, G) are independent, which proves the claim.

An r-graph F is a G_{G}-chain if every E ∈ E(F) is dependent on E(F) \ {E} in G_{G}, that is, for some h ∈ Z and real c’s, we have

e_{E} = h + ∑_{D∈E(F)\{E}} c_{D}e_{D}.  (56)

This is easily seen to be equivalent to the existence of h ∈ Z with supp(h) = E(F). To verify the last condition we have to find a solution (c_{E})_{E∈E(F)} with all entries non-zero of the system (55).

8.2 Count Matroids

Here we present the definition of a count matroid and establish some its properties. We generalize naturally the original definition of White and Whiteley [WW84] to obtain a considerably wider family of matroids for which we preserve the same name. For example, our count matroids admit many polynomials in n as the rank function while the original definition is confined to linear functions only. An advantage of count matroids is that they are defined in purely combinatorial terms and it is usually easy to identify their independent sets and circuits.

Count matroids are helpful in computing the w-sat-function, as is described in Section 9. We hope that they will have many other interesting applications; one is presented by Whiteley [Whi89].

8.2.1 Definitions

A function ρ : X^{(<∞)} → ℝ (from finite subsets of X to the reals) is called integral if it is integer-valued, increasing if ρ(A) ≤ ρ(B) whenever A ⊂ B and submodular if

ρ(A ∪ B) + ρ(A ∩ B) ≤ ρ(A) + ρ(B), A, B ∈ X^{(<∞)}.  (57)

Given ρ : X^{(<∞)} → ℝ, we say that non-empty A ⊂ X is ρ-balanced (or just balanced if ρ is understood) if |A| ≥ ρ(A) + 1 but, for every proper B ⊂ A (that is B ≠ ∅ and B ≠ A), we have |B| ≤ ρ(B).

Edmonds and Rota [ER66] observed the following result. (The proof is easy and can be found, for example, in Oxley [Oxl92, Proposition 12.1.1].)

Lemma 29 For any integral increasing submodular function ρ : X^{(<∞)} → ℝ, the family of ρ-balanced sets satisfies the circuit axioms and therefore defines a matroid on X.

We are interested in defining a matroid on X = [n]^{(r)}. (Then 2^{X} is identified with the set of r-graphs on [n].) White and Whiteley [WW84], see also [Whi96], introduced a family of count matroids on [n]^{(r)} by defining

ρ(H) = a_{1}| ∪_{E∈H} E| + a_{0}, H ⊂ [n]^{(r)},

for some fixed a_{1} and a_{0}.

We have found it possible to generalize this construction in the following way. For H ⊂ [n]^{(r)}, we denote p_{i}(H) = |∂_{i}H|, where

∂_{i}H = {D ∈ [n]^{(i)} : D ⊂ E for some E ∈ H}, i ∈ [0, r].

For example, p_{r}(H) = e(H) and p_{1}(H) = | ∪_{E∈H} E|.

We consider linear functions, that is, functions defined by

L(H) = a_{0} + ∑_{i=1}^{r−1} a_{i}p_{i}(H), H ⊂ [n]^{(r)},  (58)

for some constants a_{i} ∈ ℝ, i ∈ [0, r − 1].

Let us see when the function L satisfies the above properties for X = ℕ^{(r)}. It is easy to see that L is integral if and only if all coefficients are integers. Submodular and increasing linear functions are characterized by the following two lemmas which are of independent interest.

Lemma 30 A linear function L : X^{(<∞)} → ℝ is increasing if and only if

∑_{j=i}^{r−1} a_{j}\binom{r}{j} ≥ 0,  i ∈ [r − 1].  (59)

Proof. Suppose that L is increasing. Given i ∈ [r − 1], consider the r-graph H = {E ∈ [n]^{(r)} : |E ∩ [r]| < i}, n ≥ 2r − i + 1. We must have

L(H ∪ \{[r]\}) − L(H) = ∑_{j=i}^{r−1} a_{j}\binom{r}{j} ≥ 0,

which is exactly inequality (59).

On the other hand, suppose that L satisfies (59). Clearly, it is enough to show that, for any finite H ⊂ X and E ∈ X \ H, we have L(H) ≤ L(H ∪ {E}). Let C_{i} = ∂_{i}(H) ∩ E^{(i)}, c_{i} = |C_{i}|/\binom{r}{i}, D_{i} = E^{(i)} \ ∂_{i}(H) and d_{i} = |D_{i}|/\binom{r}{i}, i ∈ [r − 1]. Clearly, for any i and j, 1 ≤ i < j ≤ r − 1, the set system D_{i} ∪ C_{j} is an antichain in 2^{E}. By the LYM inequality, d_{i} ≤ 1 − c_{j} = d_{j}, that is,

0 ≤ d_{1} ≤ … ≤ d_{r−1} ≤ 1.  (60)

It is easy to check that

L(H ∪ \{E\}) − L(H) = ∑_{i=1}^{r−1} a_{i}d_{i}\binom{r}{i}.  (61)

Consider the problem of minimizing (61) given only the constraints (60). A moment’s thought reveals that there exists i ∈ [0, r − 1] such that the extremum is achieved when d_{1} = … = d_{i} = 0 and d_{i+1} = … = d_{r−1} = 1. But then (61) is non-negative by (59), so L is increasing.

Lemma 31 A linear function L : X^{(<∞)} → ℝ is submodular if and only if a_{i} ≥ 0, i ∈ [r − 1].

Proof. The trivial consideration shows that, for any i ∈ [r] and H, G ⊂ [n]^{(r)}, we have p_{i}(H) + p_{i}(G) ≥ p_{i}(H ∪ G) + p_{i}(H ∩ G). This implies (57) if every coefficient of L (except perhaps a_{0}) is non-negative.

On the other hand, suppose that L is submodular. Given any i ∈ [r − 1] consider the following set systems. Choose a ‘large’ m-set Z ⊂ ℕ and (r−i)-sets D_{Y} and E_{Y}, indexed by Y ∈ Z^{(i)}, so that all 2\binom{m}{i} + 1 selected sets are disjoint. Let

H = {D_{Y} ∪ Y : Y ∈ Z^{(i)}}
G = {E_{Y} ∪ Y : Y ∈ Z^{(i)}}.

Clearly, we have p_{j}(H ∩ G) = 0, j ∈ [r − 1], as H ∩ G = ∅, and

p_{j}(H) = p_{j}(G) = \{ \begin{matrix}\binom{m}{i}\binom{r}{j}, & i < j ≤ r − 1, \\ \binom{m}{i} (\binom{r}{j} − \binom{i}{j}) + \binom{m}{j}, & 1 ≤ j ≤ i, \end{matrix}
p_{j}(H ∪ G) = \{ \begin{matrix}2\binom{m}{i}\binom{r}{j}, & i < j ≤ r − 1, \\ 2\binom{m}{i} (\binom{r}{j} − \binom{i}{j}) + \binom{m}{j}, & 1 ≤ j ≤ i. \end{matrix}

Routine calculations show that

L(H) + L(G) − L(H ∪ G) − L(H ∩ G) = a_{i}\binom{m}{i} + O(m^{i−1}),

which, by the submodularity of L, implies a_{i} ≥ 0.

Thus we restrict our attention to integer coefficients satisfying

a_{i} ≥ 0, i ∈ [r − 1], and ∑_{j=0}^{r−1} a_{j}\binom{r}{j} ≥ 1,  (62)

in which case, by Lemma 29, L defines a matroid N_{L}^{n} on [n]^{(r)}, n ≥ r, which we still call a count matroid. The second condition in (62) excludes the degenerate case when already a single edge is dependent. Obviously, N_{L}^{n} is a symmetric matroid, that is, for any permutation σ of the vertex set [n], H ⊂ [n]^{(r)} is independent if and only if σ′(H) is, where σ′ denotes the induced action on [n]^{(r)}. Clearly, the nested sequence (N_{L}^{n})_{n≥r} is compatible so we do not usually specify n.

Actually, N_{L} admits an alternative definition if a_{0} ≥ 0. Let X = [n]^{(r)} and let Y be the disjoint union of a_{i} copies of [n]^{(i)}, i ∈ [0, r−1]. Define the bipartite graph G on X ∪ Y by connecting E ∈ X to all elements of Y corresponding to subsets of E ∈ [n]^{(r)}. (For example, every vertex in X has degree ∑_{i=0}^{r−1} a_{i}\binom{r}{i}.) It is easy to see that the transversal matroid of G, in which H ⊂ X is independent if and only if H can be matched into Y, equals N_{L}^{n}.

Any transversal matroid is representable over fields of every characteristics, see Piff and Welsh [PW70]; this applies to all count matroids with a_{0} ≥ 0. We do not know if N_{L} is representable for a_{0} < 0.

8.2.2 Rank

Let us determine the rank of N_{L}^{n}.

Theorem 32 Let L satisfy (62). Then R(N_{L}^{n}) = min (\binom{n}{r}, L([n]^{(r)})).

Proof. We may assume that N = N_{L}^{n} contains a non-trivial circuit for otherwise R(N) = \binom{n}{r} ≤ L([n]^{(r)}) and our claim is true.

Let an r-graph G form a base for N.

Claim 1 There exists an ordering of \overline{G} = \{E_{1}, …, E_{s}\} such that

F_{[j−1]} ∩ F_{j} ≠ ∅, j ∈ [2, s],  (63)

where F_{i} denotes the (unique and, by (62), non-empty) subgraph of G such that F_{i} + E_{i} is a circuit. (Also we denote F_{I} = ∪_{i∈I}F_{i}, F + E = F ∪ {E}, etc.)

To show the claim choose arbitrary E_{1} ∈ \overline{G} and, inductively, take for E_{j} any available edge satisfying (63). Suppose, on the contrary, that we are stuck after having chosen E_{1}, …, E_{j−1}, some j ∈ [2, s]. Let G_{1} = F_{[j−1]} and G_{2} = G \ G_{1}. Both G_{1} and G_{2} are non-empty. Clearly, for any E ∈ \overline{G} we must have either F ⊂ G_{1} or F ⊂ G_{2} where F + E is the circuit with F ⊂ G. Thus, if H_{i} is the closure of G_{i}, i = 1, 2, then H_{1} = G_{1} + E_{[j−1]} and H_{2} = [n]^{(r)} \ H_{1}.

Let C be any N -circuit. We claim that C cannot intersect both H_{1} and H_{2}. Suppose not. Let E ∈ C ∩ H_{1}. As G_{2} spans H_{2}, the rank of (C ∩ H_{1}) ∪ G_{2} wo not decrease if we remove E. Therefore, there is a circuit C′ ∋ E such that C′ ∩ H_{1} ⊂ C and C′ ∩ H_{2} ⊂ G_{2}. Likewise, fixing some D ∈ C′ ∩ G_{2} ≠ ∅, we obtain a circuit C′′ ⊂ G which contradicts the independence of G.

Note that if we replace C by the r-graph C′ composed of the first e(C) elements of [n]^{(r)} in the colex order, then p_{i}(C) will not increase by the Kruskal-Katona Theorem [Kru63, Kat66], so e(C′) > L(C′). If C′ is not a circuit, take any proper subcircuit and repeat. The first two edges, [r] and [2, r + 1], of the eventual circuit C′ (which by (62) has size at least 2) share r−1 vertices and fall into the same half of [n]^{(r)} = H_{1} ∪ H_{2}. But every two edges can be connected by a sequence of edges such that any two neighbours share r − 1 vertices. By the symmetry of N, one of the halves must be empty, which is a contradiction proving Claim 1.

Choose an ordering guaranteed by Claim 1. Let us prove, by induction on j, the following. Claim 2 L(F_{[j]} + E_{[j]}) = L(F_{[j]}) = e(F_{[j]}), j ∈ [s].

First we note that, for every i ∈ [s],

e(F_{i}) ≤ L(F_{i}) ≤ L(F_{i} + E_{i}) ≤ e(F_{i} + E_{i}) − 1 = e(F_{i}),

which implies L(F_{i} + E_{i}) = L(F_{i}) = e(F_{i}); in particular, our claim is true for j = 1. Now we argue as follows:

L(F_{[j]} + E_{[j]}) ≤ L(F_{[j−1]} + E_{[j−1]}) + L(F_{j} + E_{j}) − L(F_{[j−1]} ∩ F_{j})

≤ e(F_{[j−1]}) + e(F_{j}) − e(F_{[j−1]} ∩ F_{j}) = e(F_{[j]}).

In the above transformations, we use the submodularity of L, induction and the inequality L(F_{[j−1]} ∩ F_{j}) ≥ e(F_{[j−1]} ∩ F_{j}); the last inequality is valid because F_{[j−1]} ∩ F_{j} is independent and non-empty. (Actually, Claim 1 could be skipped if a_{0} ≥ 0.) Now, Claim 2 follows.

Clearly, F_{[s]} = G. Therefore, L([n]^{(r)}) = L(G) = e(G) = R(N_{L}^{n}).

Remark. Kalai [Kal90] showed that, for any symmetric matroid M on ℕ^{(r)}, R_{M}([n]^{(r)}) is a polynomial in n for all sufficiently large n and characterized all possible polynomials. Unfortunately, these are not confined to L([n]^{(r)}) with some L satisfying (62). For example, the k-hyperconnectivity matroid on ℕ^{(2)} introduced by Kalai [Kal85] gives the polynomial kn − \binom{k+1}{2}. It would be of interest to have a purely combinatorial construction (like that of a count matroid) producing every possible growth polynomial. (Matroids in [Kal90] are constructed by means of multilinear algebra.)

9 Proof Techniques

Here we present a few different methods for proving lower bounds on w-sat(n, F). Of these, C-proofs and G-proofs can be viewed as sufficient criteria for G_{n} ∈ w-SAT(n, F) to be of the minimal size. Our approach is based on gross and count matroids which are defined in Section 8.

The links with matroid theory are not surprising insofar as the definition of weak saturation suggests some kind of dependence; loosely speaking, an F - proper addition of edges corresponds to closure and the notion of a minimum weakly saturated graph resembles that of a base.

The following observation, due to Kalai [Kal85], is crucial to our work. Suppose that we have a matroid M on [n]^{(r)} and an r-graph F which is an M-chain, that is, for every embedding V (F) ⊂ [n], every edge E ∈ E(F) ⊂ [n]^{(r)} is dependent on E(F) \ {E}. Then we claim that the size of any weakly F -saturated graph G on [n] is at least R_{M}([n]^{(r)}), the rank of M. Indeed, let E_{1}, …, E_{k} be an F -proper ordering of E(\overline{G}). By the definition, for every i ∈ [k], there is an F -subgraph of G_{i} = G + E_{1} … + E_{i} containing E_{i}. Thus, E_{i} lies in the M-closure of G_{i−1}, which inductively implies that G spans [n]^{(r)} in M, and the claim follows.

Clearly, the above argument can be applied to a family F of forbidden rgraphs.

Lemma 33 (Kalai) We have

w-sat(n, F) ≥ l,  (64)

if we can find a matroid M on [n]^{(r)} such that every F ∈ F is an M-chain and R_{M}([n]^{(r)}) ≥ l.

In this case we say that we can m-prove the inequality (64). If, furthermore, M is a count matroid, a gross matroid, or a representable matroid, then (64) is said to be c-proved, g-proved, or r-proved correspondingly. Of course, if there exists G ∈ w-SAT(n, F) with e(G) = l, then G is extremal. In this case we say that F admits an m-proof for n. In the obvious way we define a c-proof, a g-proof, and an r-proof.

Given a matroid M on [n]^{(r)} and an r-graph F, let

D_{M}(F) = min_{F ⊂[n]} (e(F) − R_{M}(E(F))) ,

that is, for every embedding F ⊂ [n], we compute how many F -edges can be removed without decreasing the M-rank of E(F) and take the minimum over all embeddings F ⊂ [n]. For a family F of r-graphs, we define

D_{M}(F) = min{D_{M}(F) : F ∈ F}.  (65)

The following refinement of Lemma 33 is also useful.

Lemma 34 Suppose that, for some family F of r-graphs and a matroid M on [n]^{(r)}, every F ∈ F is an M-chain. Then,

w-sat(n, F) ≥ R_{M}([n]^{(r)}) + D_{M}(F) − 1.  (66)

Proof. As in Lemma 33, we conclude that E(G) spans [n]^{(r)} in M for any weakly F-saturated graph G on [n]. Consider the first edge E added to G. It creates some forbidden F ⊂ [n]; clearly, E(F) \ {E} ⊂ E(G). Therefore, there are D_{M}(F) − 1 edges in G which are dependent on the remaining edges and the lemma follows.

We say that (66) is m′-proved. If M is a count, gross, or representable matroid, then we respectively c′-prove, g′-prove, or r′-prove (66). If the lower bound in (66) is sharp, then we obtain an m′-proof. In the obvious way we define a c′-proof, a g′-proof, and an r′-proof.

The characteristic feature of w-sat-type problems is that, given F, it is usually fairly easy to come up with a correct example of a weakly F-saturated graph G (as a rule, there are many different extremal graphs) and the harder part is to prove that G is minimal. So, a typical problem is, given G ∈ w-SAT(n, F), to verify whether e(G) = w-sat(n, F), that is, we want to have some sufficient and/or necessary conditions that a weakly F-saturated graph G has the minimal number of edges. Even if there exists an m-proof, it is not obvious at all how to search for a suitable matroid.

However, the gross matroid of G seems a good candidate for M. If each element of F is a G_{G}-chain, then we immediately conclude that G is extremal. In this case say that the pair (F, G) admits a G-proof. Hence, the G-proof can be viewed as a sufficient criterion for G ∈ w-SAT(n, F) to be of the minimal size.

As gross matroids are representable, we have the following ‘hierarchy’ of proofs (and other implications):

G-proof ⇒ g-proof ⇒ r-proof ⇒ m-proof.

Unfortunately, gross matroids are not, in general, very easy to handle; it takes some efforts to identify their chains. Also, there are many examples of eligible pairs which do not accept a G-proof. For example, minimum weakly K_{3}^{2}-saturated graphs are trees, of which only stars produce a G-proof. Besides, G/g/g′-proofs do not provide an immediate characterization of minimum weakly saturated graphs, as usually there seems to be no easy combinatorial description of the set of bases of a gross matroid.

However, many new results are proved here using gross matroids. Let us prove one trivial lemma which, when combined with the results of Sections 11 and 12, has non-trivial consequences.

Lemma 35 Let K = lK_{r}^{r} be the union of l disjoint r-edges. Then G_{K} is the uniform matroid of rank l, that is, an r-graph F is independent in G_{K} if and only if e(F) ≤ l.

In particular, for any family F of r-graphs and for any n with \binom{n}{r} ≥ l, we can g-prove that w-sat(n, F) ≥ l, where l = min{e(F) : F ∈ F} − 1.

Proof. Let us show, by induction on l, that any r-graph H of size l is G_{K}- independent. We may assume that E = [r] is an edge in both these graphs. One can see that

det(M(K, H)) = ±α_{11} … α_{rr} det(M(K′, H′)) + (other terms),

where H′ and K′ are obtained respectively from H and K by removing E and none of the ‘other terms’ contains α_{11} … α_{rr} as a factor. By induction, we conclude that det(M(K, H)) ≠ 0, and the claim follows as the rank of G_{K} is e(K) = l.

Count matroids can be applied to w-sat-type problems in the following, slightly different, way. Suppose that, for a range of values of n, we have G_{n} ∈ w-SAT(n, F) (conjectured to be extremal) such that e(G_{n}) is a polynomial in n. Then we try to write explicitly the (unique, if it exists) count matroid N such that R_{N} ([n]^{(r)}) = e(G_{n}) and check whether each F ∈ F is an N -chain. If we succeed, then G_{n} is indeed extremal and we have a C-proof.

This approach is usually less successful than the one via gross matroids. Its weaknesses are that we must have a guess for a number of values of n and that not many polynomials are the growth polynomials of a count matroid. But still there are a few natural problems for which, of the above approaches, only count matroids produce results, e.g. for some uniform families, see Subsection 10.2.

10 Specific Classes

Here we obtain various results for certain particular forbidden families.

10.1 Pyramids

Here we compute the w-sat-function for pyramids, which includes a few interesting results as partial cases: for example, this proves formula (49) conjectured by Tuza [Tuz88, Conjecture 7].

Let t be fixed. Suppose we are given a sequence r = (r_{1}, …, r_{t}) of nonnegative integers and a sequence of disjoint sets S_{1}, …, S_{t} of sizes s = (s_{1}, …, s_{t}) such that r_{[i]} ≤ s_{[i]}, i ∈ [t]. (Dealing with sequences, we use such shortcuts as r_{I} = ∑_{i∈I} r_{i} and S_{I} = ∪_{i∈I}S_{i}, I ⊂ [t]; we also assume r_{0} = 0, S_{0} = ∅.)

The pyramid P = P (s; r) is the r-graph, r = r_{[t]}, on S = S_{[t]} such that E ∈ S^{(r)} is an edge of P if and only if, for every i ∈ [t], we have |E ∩ S_{[i]}| ≥ r_{[i]}. Of course, this condition is vacuously true for i = t.

For example, for t = 1 we have complete graphs; P (s_{1}, s_{2}; r_{1}, r_{2}) consists of those (r_{1} + r_{2})-subsets of S_{1} ∪ S_{2} which intersect S_{1} in at least r_{1} vertices. As a warning, we emphasize that pyramids are usual (not layered) r-graphs.

Without loss of generality we may assume that s_{i} ≥ r_{i}, i ∈ [t]. If some r_{i} exceeds s_{i} then, letting r′ = r except r′_{i} = s_{i} and r′_{i−1} = r_{i−1} + r_{i} − s_{i} (note that i ≥ 2 as r_{1} ≤ s_{1}), we obtain the same pyramid P′ = P. Indeed, r_{[j]}’s do not change except r′_{[i−1]} = r_{[i−1]} + r_{i} − s_{i}, so, trivially, P′ ⊂ P. On the other hand, E ∈ E(P) implies that

|E ∩ S_{[i−1]}| ≥ |E ∩ S_{[i]}| − s_{i} ≥ r_{[i]} − s_{i} = r′_{[i−1]},

and E ∈ E(P′). Iterating the step as long as possible, we prove the claim.

Likewise we can get rid of r_{i} = 0 by merging S_{i} and S_{i+1} together (or removing S_{t} if i = t).

Here we calculate w-sat(n, P) by showing that pyramids admit a G-proof. Note that we obtain the exact answer for all feasible values of the parameters n, r and s. This result appears in [Pik99a].

Let us, for any n ≥ s = s_{[t]}, provide a construction of G ∈ w-SAT(n, P). Partition [n] = A_{1} ∪ … ∪ A_{t+1} so that a_{i} = |A_{i}| = s_{i} + r_{i−1} − r_{i}, i ∈ [t]; thus

a_{t+1} = |A_{t+1}| = n − ∑_{i=1}^{t}(s_{i} + r_{i−1} − r_{i}) = n − s + r_{t}.

We also assume that our partition is consecutive, that is, in [n], any element of A_{i} comes before any element of A_{j} whenever i < j.

Let E ∈ [n]^{(r)} be an edge of G if and only if, for some i ∈ [t], we have |E ∩ A_{[i]}| > r_{[i−1]}. Equivalently, the complement of G is isomorphic to

P (a_{t+1}, …, a_{1}; r_{t}, …, r_{1}, 0),

so, for example, any r-tuple intersecting A_{1} is in E(G).

Lemma 36 G ∈ w-SAT(n, P).

Proof. Order the missing edges in any way so that the sequences

(|A_{[1]} ∩ E|, …, |A_{[t+1]} ∩ E|) , E ∈ E(\overline{G}),

are non-increasing in the lexicographic order. (Thus, we start with (0, r_{1}, …, r_{t}) and end with (0, …, 0, r).) Let us show that this ordering is P -proper. Consider the moment when we add some edge E ∈ E(\overline{G}). Let E_{i} = E ∩ A_{i+1}, i ∈ [t]. Also, let E = R_{1} ∪ … ∪ R_{t} and [n] \ E = T_{1} ∪ … ∪ T_{t+1} be the consecutive partitions with |R_{i}| = r_{i} and |T_{i}| = s_{i} − r_{i}, i ∈ [t].

Let us show that E_{[i]} ⊂ R_{[i]} and T_{[i]} ⊂ A_{[i]} \ E_{[i−1]}, i ∈ [t]. As all partitions in question are consecutive, it is enough to verify the sizes. By the definition of G, we have |E_{[i]}| = |E ∩ A_{[i+1]}| ≤ r_{[i]}. Also,

|A_{[i]} \ E_{[i−1]}| ≥ |A_{[i]}| − r_{[i−1]} = ∑_{j=1}^{i}(s_{j} + r_{j−1} − r_{j}) − r_{[i−1]} = |T_{[i]}|,

and the claim follows.

Let S_{i} = T_{i} ∪ R_{i}, i ∈ [t]. We claim that E creates a forbidden subgraph P on the set S = S_{[t]}. For every i ∈ [t], we have |E ∩ S_{i}| = |R_{i}| = r_{i}, so E ∈ E(P).

Suppose, on the contrary, that there exists D ∈ E(P) coming after E. Let us show by induction on i that, for every i ∈ [0, t], we have

D ∩ S_{[i]} = E ∩ S_{[i]} and D ∩ A_{[i+1]} = E ∩ A_{[i+1]},  (67)

which would be a contradiction to the assumption D ≠ E. As D, E ∈ E(\overline{G}) are disjoint from A_{1}, the claim is true for i = 0. Let i ∈ [t]. As T_{[i]} ⊂ A_{[i]}, we conclude, by the inductive assumption, that D ∩ T_{[i]} = E ∩ T_{[i]} = ∅. As S_{[i]} = T_{[i]} ∪ R_{[i]}, we have D ∩ S_{[i]} ⊂ R_{[i]}. On the other hand, D ∈ E(P) so |D ∩ S_{[i]}| ≥ r_{[i]}, which implies

D ∩ S_{[i]} = R_{[i]} = E ∩ S_{[i]},

and the first part of (67) is proved. Now,

D ∩ A_{[i+1]} ⊃ R_{[i]} ∩ A_{[i+1]} ⊃ E_{[i]} ∩ A_{[i+1]}.

By induction, D ∩ A_{[i]} = E ∩ A_{[i]} and, as D was added later than E, we must have |D ∩ A_{[i+1]}| ≤ |E ∩ A_{[i+1]}|, which proves (67) completely.

Theorem 37 The pair (P, G) admits a G-proof.

Proof. We have to show that P is a G_{G}-chain. Let us consider

h = h_{1} ∧ … ∧ h_{t}, where h_{i} = f_{A_{[i]}}^{∗} ⌞ e_{S_{[i]}} ∈ ⋀^{r_{i}}V, i ∈ [t],

where, as usual, f^{∗} is a generic V^{∗}-basis relative a V -basis e. Each E ∈ supp(h) is of the form E_{1} ∪ … ∪ E_{t}, for some E_{i} ∈ supp(h_{i}), i ∈ [t]. Clearly, |E_{i}| = r_{i} and E_{i} ⊂ S_{[i]}. Therefore, |E ∩ S_{[i]}| ≥ |E_{[i]}| = r_{[i]}, so supp(h) ⊂ E(P). Similarly, supp_{f} (h_{i}) lives within A_{[i+1,t+1]}, i ∈ [t], which implies that supp_{f} (h) ⊂ E(\overline{G}).

So, to prove the theorem, it is enough to show that for any E ∈ E(P) we have P_{E} = ⟨e_{E}^{∗}, h⟩ ≠ 0. To do so, we can assume that S is an initial segment in [n] and every element of S_{i} comes before every element of S_{j} whenever i < j. Furthermore, we may assume that E_{i} = E ∩ S_{i} is a final segment of S_{i}. Note that A_{[i]} ⊂ S_{[i]} ⊂ A_{[i+1]} and R_{i} = S_{[i]} \ A_{[i]} consists of the last r_{i} elements of S_{i}, i ∈ [t]. Clearly, |E| = |R|, where R = R_{[t]}, so let g : E \ R → R \ E be the order-preserving bijection.

As P_{E} is a polynomial in the α’s, to show that P_{E} ≠ 0, it is enough to demonstrate a particular example of the α’s (or f^{∗}) such that P_{E} ≠ 0. Define

f_{x}^{∗} = \{ \begin{matrix}e_{x}^{∗} + e_{g(x)}^{∗}, & x ∈ E \ R, \\ e_{x}^{∗}, & otherwise. \end{matrix}  (68)

Let i ∈ [t]. To compute h_{i}, we expand f_{A_{[i]}}^{∗} in the e^{∗}-basis by (68). Denote W_{i} = A_{[i]} \ (E \ R) and

X_{i} = {x ∈ A_{[i]} \ W_{i} : g(x) ∈ A_{[i]}},
Y_{i} = {x ∈ A_{[i]} \ W_{i} : g(x) ∉ S_{[i]}},
Z_{i} = {x ∈ A_{[i]} \ W_{i} : g(x) ∈ S_{[i]} \ A_{[i]}}.

As A_{[i]} ⊂ S_{[i]} we have a partition A_{[i]} = W_{i} ∪X_{i} ∪Y_{i} ∪Z_{i}. As f_{x}^{∗} = e_{x}^{∗} for x ∈ W_{i},

f_{A_{[i]}}^{∗} = ±f_{W_{i}}^{∗} ∧ f_{X_{i}}^{∗} ∧ f_{Y_{i}}^{∗} ∧ f_{Z_{i}}^{∗} = ±e_{W_{i}}^{∗} ∧ f_{X_{i}}^{∗} ∧ f_{Y_{i}}^{∗} ∧ f_{Z_{i}}^{∗}.

Take some x ∈ X_{i}; then g(x) ∈ W_{i}. Now, for some u^{∗} ∈ ⋀V^{∗}, we have the following representation

f_{A_{[i]}}^{∗} = f_{x}^{∗} ∧ f_{g(x)}^{∗} ∧ u^{∗} = (e_{x}^{∗} + e_{g(x)}^{∗}) ∧ e_{g(x)}^{∗} ∧ u^{∗} = e_{x}^{∗} ∧ e_{g(x)}^{∗} ∧ u^{∗},

which implies that f_{A_{[i]}}^{∗} = ±e_{W_{i}}^{∗} ∧ e_{X_{i}}^{∗} ∧ f_{Y_{i}}^{∗} ∧ f_{Z_{i}}^{∗}.

Next, consider some x ∈ Y_{i}; then g(x) ∉ S_{[i]}. For some u^{∗} ∈ ⋀V^{∗}, we have

f_{A_{[i]}}^{∗} ⌞ e_{S_{[i]}} = (u^{∗} ∧ f_{x}^{∗})⌞ e_{S_{[i]}} = (u^{∗} ∧ (e_{x}^{∗} + e_{g(x)}^{∗})) ⌞ e_{S_{[i]}} = (u^{∗} ∧ e_{x}^{∗})⌞ e_{S_{[i]}},

that is, we can replace f_{x}^{∗} by e_{x}^{∗} without affecting h_{i}. Also g(Z_{i}) ∩ A_{[i]} = ∅ and S_{[i]} \ A_{[i]} = R_{i}, so

h_{i} = ±(e_{W_{i}}^{∗} ∧ e_{X_{i}}^{∗} ∧ e_{Y_{i}}^{∗} ∧ f_{Z_{i}}^{∗})⌞ e_{S_{[i]}} = ±f_{Z_{i}}^{∗} ⌞ e_{Z_{i}∪R_{i}}.

For i ∈ [t], we have |E_{[i−1]}| ≥ |R_{[i−1]}| and one of E_{i} and R_{i} is a subset of the other, so, for each x ∈ E_{i} \ R, g(x) lies in R_{j} = S_{[j]} \ A_{[j]} and x ∈ Z_{j}, for some j ∈ [i + 1, t]. Therefore, Z_{[t]} = E \ R.

When we compute P_{E} = ±⟨e_{E}^{∗}, ∧_{i∈[t]}(f_{Z_{i}}^{∗} ⌞ e_{Z_{i}∪R_{i}})⟩ by expanding further each h_{i} in the e-basis, we obtain h as a sum of terms each of the form e_{D}, for some D ∈ [n]^{(r)}. By definition, ⟨e_{E}^{∗}, e_{D}⟩ = 0 unless E = D. Consider some x ∈ Z_{i} ⊂ E. As x ∉ R and Z_{1}, …, Z_{t} are disjoint, no element of supp(h_{j}) can contain x unless j = i. Computing h_{i}, we have for some u^{∗}

h_{i} = (u^{∗} ∧ f_{x}^{∗})⌞ e_{Z_{i}∪R_{i}} = (u^{∗} ∧ e_{g(x)}^{∗})⌞ e_{Z_{i}∪R_{i}} + (u^{∗} ∧ e_{x}^{∗})⌞ e_{Z_{i}∪R_{i}},

and no element in the e-support of the second summand can contain x. Thus we can harmlessly replace f_{x}^{∗} by e_{g(x)}^{∗}. (Clearly, this does not affect h_{j} for j ≠ i.) Now, since g(Z_{i}) ⊂ S_{[i]} \ A_{[i]} = R_{i},

P_{E} = ±⟨e_{E}^{∗}, ∧_{i∈[t]}(e_{g(Z_{i})}^{∗} ⌞ e_{Z_{i}∪R_{i}})⟩
= ±⟨e_{E}^{∗}, e_{Z_{[t]}∪R_{[t]}\g(Z_{[t]})}⟩ = ±⟨e_{E}^{∗}, e_{E}⟩ = ±1.

Thus P_{E} is non-zero and the theorem follows.

Corollary 38 Suppose that we are given two sequences s = (s_{1}, …, s_{t}) and r = (r_{1}, …, r_{t}) of integers such that s_{i} ≥ r_{i} ≥ 1, i ∈ [t]. Then, for n ≥ s_{[t]},

w-sat(n, P (s; r)) = \binom{n}{r_{[t]}} − ∑_{r′} \binom{n − s_{[t]} + r_{t}}{r′_{t}} ∏_{i∈[t−1]} \binom{s_{i+1} + r_{i} − r_{i+1}}{r′_{i}},

where the summation is taken over all sequences of non-negative integers r′ = (r′_{1}, …, r′_{t}) such that r′_{[t]} = r_{[t]} and, for any i ∈ [t − 1], r′_{[i]} ≤ r_{[i]}.

Remark. To achieve equality in Corollary 38, the edges of a weakly P -saturated graph H must form a base in G_{G}. As it is the case with G/g-proofs, there is no easy combinatorial interpretation of this condition.

Pyramids cover many interesting graphs as partial cases and Corollary 38 implies new results even for r = 2: we are able to compute the w-sat-function for P_{s,t} = P (s, t; 1, 1), the disjoint union of K_{s}^{2} and E_{t}^{2} plus all edges between them. Namely, for n ≥ s + t, s ≥ 1, t ≥ 1, we have

w-sat(n, P_{s,t}) = (s − 1)n − \binom{s}{2} + \binom{t}{2}.

As P (m; r) = K_{m}^{r}, we can compute w-sat(n, K_{m}^{r}), formula (46) here.

Observe that P (r − l + 1, l; r − l + 1, l − 1) is the only member of H_{r}(r + 1, l), which proves the formula (49) conjectured by Tuza [Tuz88, Conjecture 7].

Also, S_{m}^{r} = P (1, m − 1; 1, r − 1). Therefore, Corollary 38 directly implies that

w-sat(n, S_{m}^{r}) = \binom{n}{r} − \binom{n − k}{r} − k\binom{n − k}{r − 1}, n ≥ m > r ≥ 2,

where k = m−r+1. A complete description of all minimum weakly S_{m}^{r}-saturated graphs is available only for S_{m}^{2} when we can find a simple combinatorial proof which, fortunately, works for the following, wider, class of graphs.

A delta system D_{ml}^{r} contains l r-tuples so that the intersection of every two is equal to a fixed m-set called the centre. Thus, v(D_{ml}^{r}) = m + l(r − m).

Theorem 39 For any r > m ≥ 1 and n > m + l(r − m), w-sat(n, D_{ml}^{r}) = \binom{l}{2}.

Proof. To construct G ∈ w-SAT(n, D_{ml}^{r}), choose A ∈ [n]^{(m−1)} and distinct vertices y_{1}, …, y_{l−1} ∈ [n]\A. For each i ∈ [l−1], place into E(G) any l−i edges forming a D_{m,l−i}^{r}-graph centred at A ∪ {y_{i}} and disjoint from {y_{1}, …, y_{i−1}}.

Let us show that G ∈ w-SAT(n, D_{ml}^{r}). Repeat the following step for i = 1, …, l − 1. Suppose, we have already added to G all edges containing A and intersecting {y_{1}, …, y_{i−1}}. Observe that by now we have a D_{m,l−1}^{r}-subgraph centred at A ∪ {y_{i}}. It is not hard to check that we can properly add to G all edges containing A ∪ {y_{i}}, cf. Theorem 52.

Finally, add, in any order, the remaining edges so that |E ∩ A| is nonincreasing. Easy details are omitted.

Conversely, given G ∈ w-SAT(n, D_{ml}^{r}), define inductively A_{1}, …, A_{l−1} ⊂ V (G) as follows. For i = 1, …, l−1, consider the first edge added to G containing none of A_{1}, …, A_{i−1} as a subset. Let A_{i} be the centre of a created D_{ml}^{r}-subgraph F. For any j ∈ [i − 1], at most one edge of F can contain A_{j} because any two such edges overlap in A_{i} ∪ A_{j} which has size at least m + 1. Therefore, at least l − i edges of F belonged to the initial G. These edges contain A_{i} but none of A_{1}, …, A_{i−1}. So, e(G) ≥ (l − 1) + (l − 2) + … + 1 = \binom{l}{2}.

Remark. It is easy to read off the proof the characterization of all extremal graphs for S_{m}^{2} = D_{1,m−1}^{2} (and for some other cases): all minimum weakly S_{m}^{2}- saturated graphs can be obtained in the following way. Choose {x_{1}, …, x_{m−2}} ∈ [n]^{(m−2)}. For every i ∈ [m − 2], add any m − i − 1 edges through the vertex x_{i} not incident to x_{1}, …, x_{i−1}.

10.2 Uniform Families

Fix l, m, r ∈ ℕ with 1 ≤ l ≤ \binom{m}{r}. The uniform family H = H_{r}(m, l) is the family of all r-graphs of order m and size l. By definition, G ∈ w-SAT(n, H), n ≥ m, if we can consecutively add the missing edges so that each creates a new subgraph with at most m vertices and at least l edges.

There are quite a few papers dealing with the Turán ex-function for uniform families; we refer the reader to Griggs, Simonovits and Thomas [GST98] for references and for new recent results.

The sat-type problems for uniform families were considered by Tuza [Tuz88], who made a conjecture about the value of w-sat(n, H_{r}(r + 1, l)) (formula (49) here), and by Erdős, Füredi and Tuza [EFT91] who settled the case l = 3 of Tuza’s conjecture. Observe that we have essentially only one graph in H_{r}(r+1, l) which consists of all edges containing some fixed (r−l+1)-set. In our notation it is denoted by P (r−l+1, l; r−l+1, l−1), and Corollary 38 implies formula (49).

However, the general case is still open.

Here we present, for all sets of parameters, a construction of a weakly H_{r}(m, l)-saturated graph which we conjecture to be extremal. Our conjecture is in perfect accordance with the above results.

Clearly, our construction gives an upper bound. To establish some lower bounds, we use use gross and count matroids. This way we verify our conjecture for more sets of parameters. In certain cases, we characterize the sets of minimum weakly H-saturated graphs. In particular, we answer a question by Erdős, Füredi and Tuza [EFT91] who asked for a characterization of the extremal graphs for H_{r}(r + 1, 3). These results appear in [Pik98].

10.2.1 Construction

Let n ≥ m, 1 ≤ l ≤ \binom{m}{r} and H = H_{r}(m, l). We build, inductively on n, an example of a weakly H-saturated graph G_{n} = G(n, r, m, l) on [n]. If n = m, then we can take for G_{n} any member of H_{r}(m, l − 1). If n > m, then choose inductively any G_{n−1} = G(n − 1, r, m, l) and G′ = G(n − 1, r − 1, m − 1, l′), where l′ = l − \binom{m−1}{r}. (If l ≤ \binom{m−1}{r} + 1 then we take the empty graph for G′.) Let G_{n} be the r-graph on [n] defined by

E(G_{n}) = E(G_{n−1}) ∪ {E ∪ {n} : E ∈ G′}.

Let us show that G_{n} is indeed weakly H-saturated. By the definition of G_{n−1}, we can add edges so that [n − 1] spans the complete r-graph. Then add edges E_{1} ∪ {n}, …, E_{s} ∪ {n}, where (E_{1}, …, E_{s}) is any H_{r−1}(m − 1, l′)-proper ordering of the complement of G′. As each E_{i} creates a subgraph of size l′ on some (m − 1)-set M ⊃ E_{i}, M ∪ \{n\} ⊂ V (G) spans at least l′ + \binom{m−1}{r} = l edges after E_{i} ∪ {n} has been added, which shows that G_{n} ∈ w-SAT(n, H).

Conjecture 40 For any n, r, m, l ∈ ℕ satisfying m ≤ n and 1 ≤ l ≤ \binom{m}{r}, G(n, r, m, l) is a minimum weakly H_{r}(m, l)-saturated graph.

Remark. Generally, not all extremal graphs are given by our construction, cf. Theorem 44.

Let us compute the size of G_{n}. Given l ≥ 2, define (uniquely) c and d so that

l = c + 1 + ∑_{j=0}^{d−1} \binom{m − j − 1}{r − j}, c ∈ [\binom{m−d−1}{r−d}], d ∈ [0, r − 1].

The definition of G_{n} implies, after some thought, the following formula for e(G_{n}) which, alternatively, can be routinely checked by induction on n.

e(G_{n}) = ∑_{i=0}^{d} (c + ∑_{j=i}^{d−1} \binom{m − j − 1}{r − j}) \binom{n − m + i − 1}{i}, n ≥ m.

(We agree that \binom{i}{0} = 1, for any i.) For our purposes, we have to find a representation of the form e(G_{n}) = ∑_{k=0}^{d} a_{k}\binom{n}{k}. The substitution \binom{n−m+i−1}{i} = ∑_{k=0}^{i}(−1)^{i−k}\binom{n}{k}\binom{m−k}{i−k} which is an instance of Vandermonde’s convolution (see e.g. [GKP89, p. 174]), implies

a_{k} = ∑_{i=k}^{d}(−1)^{i−k}\binom{m − k}{i − k} (c + ∑_{j=i}^{d−1} \binom{m − j − 1}{r − j}) .

Now, occasionally applying the identity ∑_{i=0}^{t}(−1)^{i}\binom{j}{i} = (−1)^{t}\binom{j−1}{t}, t ≥ 0, we can find that a_{k} = (−1)^{d−k}c\binom{m−k−1}{d−k} + (−1)^{k}s_{k}, where

s_{k} = ∑_{j=k}^{d−1} \binom{m − j − 1}{r − j} ∑_{i=k}^{j}(−1)^{i}\binom{m − k}{i − k}
= (−1)^{d−1}\binom{m − k − 1}{r − k}\binom{r − k − 1}{d − k − 1}.

Therefore, in summary,

e(G_{n}) = ∑_{k=0}^{d}(−1)^{d−k} (c\binom{m − k − 1}{d − k} − \binom{m − k − 1}{r − k}\binom{r − k − 1}{d − k − 1}) \binom{n}{k}.

One can check that Conjecture 40 is compatible with (44), which is one more point supporting Conjecture 40.

10.2.2 Applications of Count Matroids

Recall that the size of G_{n} = G(n, r, m, l) is ∑_{k=0}^{d} a_{k}\binom{n}{k}, where

a_{k} = (−1)^{d−k} (c\binom{m − k − 1}{d − k} − \binom{m − k − 1}{r − k}\binom{r − k − 1}{d − k − 1}) .  (69)

We define L = ∑_{i=0}^{d} a_{k}p_{k}, so that L([n]^{(r)}) = e(G_{n}), the conjectured value. If L defines a matroid and every F ∈ H_{r}(m, l) is an N_{L}-circuit then we can conclude that w-sat(n, H_{r}(m, l)) = e(G_{n}), which establishes the validity of our conjecture in this case.

The condition a_{k} ≥ 0, k ∈ [d], can be rewritten as

(−1)^{d−k}c ≥ (−1)^{d−k} \begin{matrix}\binom{m−k−1}{r−k} & \binom{r−k−1}{d−k−1} \\ ( & {m−k−1 \atop d−k} & ) \end{matrix} = (−1)^{d−k} \frac{d − k}{r − k} \binom{m − d − 1}{r − d}.

The modulus of the latter expression is strictly decreasing with k, so, unfortunately, no suitable c would satisfy the conditions unless d ≤ 2 and we have to confine ourselves to the three cases below.

Case 1: d = 0. In this case the problem is trivial: it is easy to prove directly the following result (also observed by Erdős, Füredi and Tuza [EFT91]).

Lemma 41 For n ≥ m ≥ r ≥ 1 and 1 ≤ l ≤ \binom{m−1}{r} + 1,

w-sat(n, H_{r}(m, l)) = l − 1.

All extremal graphs are can be obtained by adding n − m isolated vertices to an F ∈ H_{r}(m, l − 1). (Which is exactly what our construction says.)

Case 2: d = 1. Let l = \binom{m−1}{r} + 1 + c, 1 ≤ c ≤ \binom{m−2}{r−1}. By (69), we let a_{1} = c and a_{0} = \binom{m−1}{r} − c(m − 1), that is,

L(H) = cp_{1}(H) + \binom{m − 1}{r} − c(m − 1), H ⊂ [n]^{(r)}.

The condition 1 ≤ a_{1}r + a_{0} implies that either m = r + 1 (then c ≤ \binom{m−2}{r−1} must equal 1) or m ≥ r + 2 and

c ≤ min (\frac{\binom{m−1}{r} − 1}{m − r − 1}, \binom{m − 2}{r − 1}) = \frac{\binom{m−1}{r} − 1}{m − r − 1},

which we assume.

Let us show that every F ∈ H_{r}(m, l) is a circuit in N_{L}. Obviously, p_{1}(F) = m, so e(F) = L(F) + 1 and F is not independent. Take any proper F′ ⊂ F. If p_{1}(F′) = m then L(F′) = L(F) ≥ e(F′). If p_{1}(F′) ≤ m − 1 then F′ is independent by Theorem 32 as L([m − 1]^{(r)}) = \binom{m−1}{r}. Hence F is a circuit and our conjecture is true.

Lemma 42 Given r, m, l and n with n ≥ m > r ≥ 2, let c = l − \binom{m−1}{r} − 1. If m > r + 1 and 1 ≤ c < \frac{1}{m−r−1} \binom{m−1}{r} or if m = r + 1 and c = 1 (when l = 3), then w-sat(n, H_{r}(m, l)) = (l − 1) + c(n − m).

In some cases, we can characterize extremal graphs by providing a combinatorial proof.

Lemma 43 In addition to the assumptions of Lemma 42, assume that m > r+1 and c < \frac{1}{m−1} \binom{m−1}{r}. Then any minimum G ∈ w-SAT(n, H_{r}(m, l)) is given by our construction.

Proof. Let \overline{G} = \{E_{1}, …, E_{s}\} be a proper ordering; suppose that each E_{i} creates a forbidden subgraph on an m-set M_{i} ⊂ [n] and let L = a_{1}p_{1} + a_{0} be as above. We know that any A ⊂ [n] spans at most a_{1}|A| + a_{0} edges in G. (In fact, this is easy to see directly for otherwise we could replace these edges by a copy of G(|A|, r, m, l), which would produce a smaller weakly saturated graph.)

We prove by induction on i that, for any i ∈ [s], H_{i} ⊂ G, the subgraph spanned by M_{[i]} ⊂ [n], is given by our construction.

Clearly, this is the case for i = 1.

Let i > 1. We have to consider only the case when k = |M_{i} \ M_{[i−1]}| ≥ 1. Of l edges of a forbidden subgraph F created by E_{i}, at most \binom{m−k}{r} can belong to H_{i−1}, which shows that

e(H_{i}) − e(H_{i−1}) ≥ l − \binom{m−k}{r} − 1 = c + \binom{m−1}{r} − \binom{m−k}{r}.

It is routine to check that the last expression is strictly greater than ck for k ∈ [2, m]. To prevent the contradiction |H_{i}| > a_{1}|M_{[i]}| + a_{0}, we must have k = 1 and E_{i} \ M_{[i−1]} = {x} for some vertex x contained in exactly c edges of F ∩G. These edges (minus x) must lie within the (m−1)-set M_{[i−1]} ∩M_{i}, which is exactly what our construction says.

As we mentioned, the value of w-sat(n, H_{r}(r+1, 3)) was computed by Erdős, Füredi and Tuza [EFT91]. They asked if there is a characterization of the extremal graphs. Our Lemma 43 does not cover this case but we can provide a different proof of the lower bound which gives us the desired characterization. Some ideas from [EFT91] are used here but, of course, we have to be more delicate if we want to extract the cases of equality.

Theorem 44 For H = H_{r}(r + 1, 3) we have

w-sat(n, H) = n − r + 1,  n ≥ r.  (70)

Every extremal graph G can be obtained in the following way. Start with the set system G containing only one edge [n]. As long as possible, remove from G any edge E of size at least r + 1, choose A ∈ E^{(r−1)}, partition E \ A = X_{1} ∪ X_{2}, X_{1}, X_{2} ≠ ∅, and add to G the edges A ∪ X_{1} and A ∪ X_{2}.

Proof. Although we have already established (70), we have to provide a combinatorial proof of the lower bound. Let G ∈ w-SAT(n, H). Note that every vertex in G is covered by at least one edge because otherwise the first edge added to G and containing this vertex cannot create a forbidden subgraph.

Let E_{1}, …, E_{j} be the edges of G. With this sequence we do, step by step and as long as possible, the following operation. If some 2 sets have at least r − 1 common points we merge them together, that is, replace them by their union (so the resulting system is no longer r-uniform).

We claim that we end up with a sequence containing a single member (which then must be equal to V (G)). Suppose not. Let Y_{1}, …, Y_{t}, t ≥ 2, be the eventual family. Every two different resulting sets can have at most r −2 common points. Obviously, every edge of G lies within some Y_{i}. Let E ∈ \overline{G} be the first edge added to G which does not lie entirely within some Y_{i}. (If for every E ∈ [n]^{(r)} there is Y_{i} ⊃ E, then, considering chains of r-sets with overlaps of size r − 1, we conclude that Y_{i} = [n], some i.) The addition of E must have created F ∈ H. The two other edges E_{1}, E_{2} ∈ E(F) either belong to G or were added before E and share r − 1 vertices, so they lie each within some set Y_{i}. But then Y_{i} must contain E ⊂ E_{1} ∪ E_{2} which is a contradiction. The claim is proved.

Now it is easy to prove by induction that in the above process every set of size m was a merger of at least m − r + 1 edges of G. Trivially, it was the case for all initial sets which were precisely the edges of G. If we merge together 2 sets of sizes m_{1} and m_{2} made of e_{1} ≥ m_{1} − r + 1 and e_{2} ≥ m_{2} − r + 1 Gedges respectively, the resulting set has at most m_{1} + m_{2} − r + 1 vertices and e_{1} + e_{2} ≥ m_{1} + m_{2} − 2r + 2 edges produced it, so the claim follows by induction.

If we have equality in (70), then, in each step of the merging procedure, every two sets merged together have exactly r − 1 common vertices, so every extremal graph can be obtained by reversing the merging process described in the statement of the theorem (of course in many different ways, generally).

We have to show that any anti-merging produces an extremal graph. Clearly, at the end we are left with r-subsets and we have exactly n − r + 1 of these. To complete the theorem, it is enough to show that a union of two complete rgraphs H_{1} and H_{2} of order at least r each with intersection A = V (H_{1}) ∩ V (H_{2}) of size r − 1, is weakly S-saturated. But this is easy: for i = r − 2, r − 1, …, add the missing edges which intersect A in exactly i points.

Remark. The construction of G(n, r, r + 1, 3) before Conjecture 40 does not cover all cases as is demonstrated, for example, by r = 3, n = 6 and

G = {{1, 2, 3}, {2, 3, 4}, {4, 5, 6}, {5, 6, 1}}.

Case 3: d = 2. Assume r ≥ 3 and l = \binom{m−1}{r}+\binom{m−2}{r−1}+c+1 with c ∈ [\binom{m−3}{r−2}]. By (69), we let a_{2} = c, a_{1} = −c(m − 2) + \binom{m−2}{r−1} and a_{0} = c\binom{m−1}{2} − (r − 1)\binom{m−1}{r}.

Let us check when L satisfies (62). Of course, a_{2} ≥ 1. Next, the condition a_{1} ≥ 0 is, in our case, c ≤ \binom{m−2}{r−1}(m − 2)^{−1}. It is false for m = r + 1, so assume m ≥ r + 2. The inequality 0 < a_{2}\binom{r}{2} + a_{1}r + a_{0} reduces to

0 < c\binom{m − r − 1}{2} + (r − \frac{(m − 1)(r − 1)}{r}) \binom{m − 2}{r − 1}.  (71)

Note that (71) is automatically true if m = r + 2 (when the coefficient at c is zero), but then the condition a_{1} ≥ 0 implies c = 1. So, we conclude that L satisfies (62) if and only if either m = r + 2 and c = 1 or m ≥ r + 3 and

\frac{((m − 1)(r − 1) − r^{2})\binom{m−2}{r−1}}{r\binom{m−r−1}{2}} < c ≤ min (\frac{\binom{m−2}{r−1}}{m − 2}, \binom{m − 3}{r − 2}) = \frac{\binom{m−2}{r−1}}{m − 2}. (72)

Let us check that any F ∈ H_{r}(m, l) is a circuit in N_{L}. Clearly, every two vertices in F are covered by an edge for otherwise we would have at most \binom{m}{r} − \binom{m−2}{r−2} < l edges in F. Therefore, L(F) = L([m]^{(2)}) = l − 1 = e(F) − 1 and we conclude that F is not N_{L}-independent. On the contrary suppose that L(H) < e(H) for some r-graph H on [m] with at most l − 1 edges. Clearly, we may assume that H is an initial segment of [m]^{(r)} in the colex order.

Note that L([m − 1]^{(r)}) = \binom{m−1}{r} and, by Theorem 32, [m − 1]^{(r)} is independent. Therefore, H must have m vertices. Also the 2-set {m − 1, m} cannot be covered by an H-edge, as then e(H) ≥ L([m]^{(r)}) + 1 ≥ l. Let H′ be the (r − 1)-graph on [m − 2] satisfying

E(H) = [m − 1]^{(r)} ∪ {D ∪ {m} : D ∈ E(H′)}.

If we let L′ = a_{2}p_{1} + a_{1} then L′([m − 2]^{(r−1)}) = \binom{m−2}{r−1} and, by Theorem 32, H′ ⊂ [m − 2]^{(r−1)} is independent in N_{L′} and L′(H′) ≥ e(H′).

Obviously, p_{2}(H) = p_{1}(H′) + \binom{m−1}{2}. Therefore,

L(H) = L([m − 1]^{(r)}) + L′(H′) ≥ \binom{m − 1}{r} + e(H′) = e(H),

which is the desired contradiction.

Theorem 45 Assume that r ≥ 3 and l = \binom{m−1}{r} + \binom{m−2}{r−1} + c + 1 are such that either m = r+2 and c = 1 or m ≥ r+3 and c satisfies (72). Then Conjecture 40 is true.

Remark. Unfortunately, we do not have any characterization of the extremal graphs in this case.

10.2.3 Applications of Gross Matroids

We establish some further results by applying gross matroids. Namely, we prove that our conjecture is asymptotically true for d = r − 1. Moreover, by applying the g′-method we settle completely the case r = 2.

First, we need one simple preliminary result.

Lemma 46 Let G be an r-graph of order n and size at least \binom{n}{r} − n + m, where n > m > r ≥ 2. Then any E ∈ E(G) is contained in a complete subgraph of order m.

Proof. Given E ∈ E(G), remove from each missing edge one (arbitrary) vertex not belonging to E. We are left with at least m vertices spanning a complete subgraph which contains E.

Remark. The above bound on e(G) is sharp: if the complement of G consists of n − m + 1 edges containing some fixed (r − 1)-set A then this set is covered only by m − r G-edges of which none lies within K_{m}^{r}.

Theorem 47 Let l = \binom{m}{r} − k and H = H_{r}(m, l). If m > k + r, then

w-sat(n, H) = (m − k − r)\binom{n}{r − 1} + O(n^{r−2}).  (73)

Furthermore, if r = 2, then we have a g′-proof that

w-sat(n, H_{2}(m, l)) = (m − k − 2)(n − m) + l − 1, n ≥ m.  (74)

Proof. Implementing our construction, from the identity ∑_{i=0}^{r} \binom{m−i−1}{r−i} = \binom{m}{r}, we obtain that d = r − 1 and c = m − r − k, which implies the upper bounds in (73) and (74).

On the other hand, in any F ∈ H, any edge lies within a K_{m−k}^{r}-subgraph by Lemma 46. But by Theorem 37, K_{m−k}^{r} is a chain in G_{P}, the gross matroid of P = P (c, n − c; 1, r − 1), so each F ∈ H is a G_{P} -chain. By Lemma 33, w-sat(n, H) ≥ R_{G_{P}} ([n]^{(r)}) = e(P), which g-proves the required lower bound in (73).

Finally, let us g′-prove the lower bound in (74) for r = 2. Let F ∈ H. As F has m vertices,

R_{G_{P}} (F) ≤ R_{G_{P}} (K_{m}) ≤ e(P (c, m − c; 1, 1)) = cm − \binom{c+1}{2}.

(The second inequality is true because P (c, m − c; 1, 1) ∈ w-SAT(m, K_{m−k}^{2}) and K_{m−k}^{2} is a G_{P} -chain.) Therefore some set of at least p = l−cm+\binom{c+1}{2} edges of F lies in the G_{P} -span of the remaining edges, that is, D_{G_{P}} (F) ≥ p. By Lemma 34,

w-sat(n, H) ≥ R_{G_{P}} (K_{n}) + D_{G_{P}} (F) − 1
≥ cn − \binom{c + 1}{2} + p − 1 = c(n − m) + l − 1.

The theorem is proved.

Note that for r = 2 we know w-sat(n, H_{2}(m, l)) for any any feasible m and l: for l ≤ \binom{m−1}{2} we have a g-proof that it is l − 1 (constant) by Lemma 35, while all other cases are covered by the g′-proof of Theorem 47.

Also note that, under the assumptions of Theorem 47 on l, the graph G(n, r, m, l) constructed before Conjecture 40 is weakly F -saturated, where E(F) consists of the first l elements of [m]^{(r)} in the colex order. So, Theorem 47 remains valid if F is the only member of H; this covers all possible cases for r = 2 except the trivial case l = \binom{m−1}{2} + 1.

10.3 Miscellaneous Graphs

Here we indicate a few easy results for some simple forbidden graphs such as cycles, disjoint edges, trees, etc. The proofs are easy but they often require a lengthy and boring verification that the specified graph is weakly saturated. We include them for the sake of completeness.

Cycles

Let C_{l} denote the cycle of length l. We know (see Section 2) that the determination of the exact value of sat(n, C_{l}) is a hard task. For the w-sat-function, on the contrary, the complete answer is available in all cases.

The following trivial observation will be used a few times, so we state it as a lemma.

Lemma 48 Let l ≥ 4 be even. Then any weakly C_{l}-saturated graph G contains an odd cycle.

Proof. Indeed, otherwise G is a bipartite graph. Let E be the first added edge lying within one part. By the parity argument, any l-cycle through E must contain another edge lying within a part, which is a contradiction to the choice of E.

Let us first consider the case when the forbidden cycle is Hamiltonian.

Theorem 49 For any n ≥ 4, w-sat(n, C_{n}) = n and all extremal graphs are obtained from a Hamiltonian cycle by adding an edge which creates an odd cycle and then removing some other edge.

Proof. Let G be a Hamiltonian cycle visiting the vertices 1, 2, …, n ∈ [n] in this order, minus the edge {1, n} but plus the edge {i, n}, for some even i. To prove that G ∈ w-SAT(n, C_{n}) we have to show how to properly add the missing edges to G. First we add {1, n} thus creating a Hamiltonian cycle through 1, 2, …, n.

We fix this cycle and define a t-chord as an edge connecting 2 vertices at a distance t if we go along the cycle. Thus, after the first step, G is made of all 1-chords and one i-chord. Next, we add all i-chords in the following order {m, i + m}, m = 1, 2, …, n − 1. (Of course, we do all arithmetic modulo n.) Every time we receive an extra cycle: for example, the chord {1, i + 1} creates the cycle via

n, i, i − 1, i − 2, …, 2, 1, i + 1, i + 2, …, n − 2, n − 1.

Having all chords of length 1 and i ≥ 4, it is possible to add any (i−2)-chord. For example, the chord {2, i} creates the following Hamiltonian cycle

1, i + 1, i + 2, 2, i, i − 1, …, 4, 3, i + 3, i + 4, …, n − 1, n.

Therefore, we can eventually have all 2-chords.

Finally, consequently for m = 3, 4, 5, …, we add all missing m-chords in any order. This is legitimate; when we add, for example, the chord {1, m + 1} we have a Hamiltonian cycle via

1, 2, …, m − 1, n, n − 1, …, m + 2, m, m + 1

which uses only already present chords (of length 1, 2 and m − 1). Therefore, G ∈ w-sat(n, C_{n}).

On the other hand, suppose that G ∈ w-SAT(n, C_{n}). The first edge added to G creates a C_{n}-subgraph F (that is, a Hamiltonian cycle), so there is a Hamiltonian path P_{n−1} in G. It is easy to see that P_{n−1} ∉ w-SAT(n, C_{n}) so there is at least one more edge E and w-sat(n, C_{n}) ≥ n. Moreover, F + E must contain at least one odd cycle by Lemma 48, which is precisely what our construction says.

Let us consider odd and even cycles separately.

Theorem 50 Let l ≥ 3 be odd and let n > l. Then w-sat(n, C_{l}) = n − 1, all extremal graphs are trees of order n and diameter at least l − 1, and C_{l} admits a g-proof for n.

Proof. Let G be any such tree. First we add any edge connecting two vertices at distance l − 1; suppose the created l-cycle goes through the vertices 1, …, l ∈ V (G) in this order. As v(G) > l and G is connected, we may assume that the vertex l + 1 ∈ V (G) is connected to l. Obviously, we can add the edge {2, l + 1} which creates the l-cycle through 2, 3, …, l, l + 1. Next, we can add the edge {1, 4} which creates the l-cycle through 4, 5, …, l, l + 1, 2, 1. Now the set [l] ⊂ V (G) spans an l-cycle plus the edge {1, 4} creating an odd (l − 2)- cycle—the situation in which we can apply Theorem 49 to add all edges within [l].

But it is trivial to show that a connected graph with an l-clique is weakly C_{l}-saturated, which implies G ∈ w-SAT(n, C_{l}).

Now it is easy to deduce that w-SAT(n, C_{l}) consists exactly of all connected graphs containing a path of length l − 1 as a subgraph and the desired characterization of the minimum ones follows.

Let us show that C_{l} admit a g-proof for n > l. Indeed, consider G = G_{S_{n}^{2}}. Any edge of C_{l} is G-dependent on the remaining ones because the path with l edges is weakly K_{3}^{2}-saturated and K_{3}^{2} is a G-circuit. Clearly, R_{G}([n]^{(2)}) = n − 1. (In fact, if restricted to [n]^{(2)}, G is the usual cycle matroid.) The claim clearly follows.

Theorem 51 Let l ≥ 4 be even and let n ≥ l. Then w-sat(n, C_{l}) = n, all extremal graphs are trees of order n and diameter at least l − 1 plus an extra edge creating an odd cycle, and C_{l} admits an r-proof for n.

Proof. Similarly to the proof of Theorem 50, to show that any indicated graph G is weakly C_{l}-saturated, we first argue that adding a few edges we can obtain an l-cycle containing a 3-chord. Unfortunately, this configuration is not weakly C_{l}-saturated but, like in Theorem 49, we can add all 3-chords, 5-chords, and so on to obtain the complete bipartite graph K_{l/2,l/2}.

Observe that having an edge {x, y} with y belonging be K_{s,t}-subgraph with s, t ≥ l/2, we can connect x to any vertex lying in the same part as y. Hence, we can add edges so that G contains a K_{s,n−s}-subgraph with s, n − s ≥ l/2; moreover, as we have an odd cycle present in the original G, one part spans an edge and G ∈ w-SAT(n, C_{l}).

The required characterization of extremal graphs easily follows.

Finally, let M be Doob’s [Doo73] even-cycle matroid on [n]^{(2)} which can be represented by f : [n]^{(2)} → V which maps {i, j} to e_{i} + e_{j} for some basis {e_{1}, …, e_{n}} of a real vector space V. The cycle C_{l} is an M-chain: if C_{l} goes via the vertices 1, 2, …, l, 1, then we have the linear relation

(e_{1} + e_{2}) − (e_{2} + e_{3}) + … − (e_{l} + e_{1}) = 0

with all coefficients non-zero.

For n ≥ 3, the rank of M is n as any basis vector e_{i} admits a representation e_{i} = \frac{1}{2}((e_{i} + e_{j}) + (e_{i} + e_{k}) − (e_{j} + e_{k})), which implies our claim.

Remark. Probably, even cycles do not admit a g-proof. But if we consider (1, 1)-layered (i.e. bipartite) graphs, then C_{4} as the complete (1, 1)-graph admits a G-proof by the results of Section 12. A little more work shows that any even cycle admits a g-proof in the class of bipartite graphs, because any connected (1, 1)-graph is weakly C_{4}-saturated.

Disjoint Edges

Suppose that we forbid lK_{r}^{r}, l > 1, that is, l disjoint r-edges.

Theorem 52 Let F = lK_{r}^{r}, let n > lr, and let G consist of l − 1 disjoint redges plus n − r(l − 1) isolated vertices. Then w-sat(n, F) = l − 1, G is the only extremal graph, and the pair (F, G) admits a G-proof.

Proof. Let us show that G is weakly F -saturated. As v(G) > kl, we can add an edge disjoint from the edges of G which creates a copy of F and leaves at least one vertex of G isolated.

Fix any D ∈ [n]^{(r)}. We have to show that D ∈ Cl_{F} (G). We prove that the existence of E ∈ Cl_{F} (G) with |E ∩D| = k < r implies that there is E′ ∈ Cl_{F} (G) with |E′ ∩ D| = k + 1. Given E, there are E_{2}, …, E_{l} ∈ Cl_{F} (G) which together with E form an F -subgraph. If there is x ∈ D \ V, V = E ∪ E_{[2,l]}, then we can take E′ = E + x − y ∈ Cl_{F} (G), for some y ∈ E \ D. Otherwise take any x ∈ D\E, say x ∈ E_{2}, replace E_{2} by E′_{2} = E_{2}−x+y ∈ Cl_{F} (G), where y ∉ V, and consider E′ = E − z + x, z ∈ E \ D which (together with E′_{2}, E_{3}, …, E_{l}) creates a forbidden subgraph. The required E′ is found. Hence, w-sat(n, F) ≤ l − 1.

Any weakly F -saturated graph contains l − 1 disjoint edges; hence G is the only extremal graph.

The pair (F, G) admits a G-proof by Lemma 35.

However, if the forbidden graph is a perfect matching, then the exact answer is known generally for r = 2 only.

Theorem 53 For n = 2l ≥ 4, w-sat(n, lK_{2}^{2}) = n − 1 and all extremal graphs can be obtained in the following way: complete lK_{2}^{2} to a tree T, add an edge E creating an odd cycle and remove any edge E′ contained in some perfect matching of T + E.

Proof. Let us show that any above constructed graph G is weakly F -saturated, F = lK_{2}^{2}. First we add the edge E′. Let C be the odd cycle (at this stage it is unique) of the obtained graph T′ = G + E′ = T + E and let M be some matching of T′. Let {x_{1}, y_{1}}, …, {x_{2k+1}, y_{2k+1}} be all edges of M with x_{i} ∈ C and y_{i} ∉ C. Claim 1 A disjoint union of an odd cycle C_{2p−1} an a even path P_{2q} is weakly (p + q)K_{2}^{2}-saturated, p ≥ 2, q ≥ 0.

We prove the claim by induction on q. If q = 0 then we can first connect the isolated vertex to any other vertex of the cycle to obtain a wheel and then we can add the remaining edges in any order. If q > 0 then we can connect the endpoints of the path to all vertices on a cycle and the obtained graph is easily seen to contain C_{2p+1} ⊔ P_{2q−2} and the claim follows.

A moment’s thought reveals that, by Claim 1, T′ ∈ w-SAT(n, F) if k = 0. So, to prove that G ∈ w-SAT(n, F), we show that, for k > 0, we can F -properly add some extra edges to T′ and find other, strictly larger, odd cycle C.

Assume that x_{1}, …, x_{2k+1} lie on the cycle C in this order clockwise. Note that we can add to T′ all edges of the form {y_{i}, y_{i+1}}, i ∈ [2k + 1], which creates the matching M′ = M△C_{i}, where C_{i} the cycle via y_{i+1}, y_{i}, x_{i}Cx_{i+1}, y_{i+1} created just now. (By aCb we denote the part of the cycle C going clockwise from a to b inclusive.) If there are no vertices (along C) between some x_{i} and x_{i+1} then we have a strictly longer cycle C△C_{i} as desired. Otherwise, we may assume that a part of the cycle C looks like x_{1}, …, a, x_{2}, b, …, c, x_{3}, . . .. It is routine to check that the addition of the edge {a, c} creates a matching which uses edges {x_{2}, b}, {y_{1}, y_{2}}, {x_{i}, y_{i}}, i ∈ [3, 2k + 1], etc. But then we can find a strictly longer odd cycle: x_{3}Ca, c C^{−1}x_{2}, y_{2}, y_{3}, x_{3}, which proves that G ∈ w-SAT(n, F) as claimed.

On the other hand, consider any weakly F -saturated graph G and let G′ = G + E be a graph with a perfect matching. If G′ is not connected, then all its components have even order, but then the first F -properly added edge not lying within a component cannot create a matching (by the parity argument), which is a contradiction. If G′ is a bipartite graph, then its parts must be of the same size, but then the first F -properly added edge lying within a part creates no perfect matching, which a contradiction. Hence, G′ is a tree plus an edge creating an odd cycle and all claims of the theorem easily follow.

Dumb-Bells

Recall that the 2-graph B_{kk}, called a dumb-bell, consists of two disjoint copies of K_{k}^{2} plus one edge connecting them, k ≥ 3.

Theorem 54 Let k ≥ 3, n = lk + q, q ∈ [0, k − 1], l ≥ 2; let ε_{q} = 1 except ε_{0} = 0. Then w-sat(n, B_{kk}) = (l + 1)\binom{k}{2} − \binom{k−q}{2} − ε_{q}.

Proof. To prove the upper bound consider the 2-graph G on [n] defined (for any q) by

E(G) = (∪_{i∈[l]}A_{i}^{(2)}) ⋃ ([n − k + 1, n]^{(2)} \ {n − 1, n}) ,

where A_{i} = [ki − k + 1, ki], i ∈ [l]. As G[kl] ≅ lK_{k}^{2}, we can add all missing edges within [kl] each connecting some two of the A’s. If q = 0, then we are done; otherwise we add the edge {n, n − 1} making G[n − k + 1, n] complete and then add the remaining edges in any order. Hence, G ∈ w-SAT(n, B_{kk}) and the upper bound follows.

On the other hand, let G ∈ w-SAT(n, B_{kk}) be arbitrary. Similarly to Lemma 43, we take a B_{kk}-proper ordering \overline{G} = \{E_{1}, …, E_{e}\}; assume that E_{i} creates a B_{kk}-subgraph F_{i} on a 2k-set M_{i} ⊂ [n]. Define the surplus s(X) = e(G[X]) − \frac{k−1}{2}|X|, X ⊂ [n], and s_{i} = s(M_{[i]}).

Let q_{i} ∈ [0, k − 1] be equal to |M_{i} \ M_{[i−1]}| (mod k). Given q_{i}, it is routine to see that if q_{i} = 0 then s_{i} ≥ s_{i−1} and if q_{i} > 0 then

s_{i} − s_{i−1} ≥ f(q_{i}) = \binom{k}{2} − \binom{k−q_{i}}{2} − \frac{k−1}{2}q_{i} − 1 ≥ 0.

Furthermore, for p, q ≥ 1, f(p + q) ≤ f(p) + f(q). Hence, s(M_{[e]}) ≥ f(q) for q > 0 and s(M_{[e]}) ≥ 0 for q = 0. Now, the identity e(G) = \frac{k−1}{2}n + s(V (G)) implies the required lower bound.

Remark. In fact, we c-prove that w-sat(n, B_{kk}) ≥ \frac{k−1}{2}n for odd k, which is sharp for n = kl. (For even k, the function L = \frac{k−1}{2}p_{1} is not integral.)

Forests

Let us consider 2-graphs. Let T be a forest of order m. Clearly, K_{m−1}^{2} plus n − m + 1 isolated vertices is weakly T -saturated, so w-sat(n, T) ≤ \binom{m−1}{2}. This is sharp for T = S_{m}^{2} by Corollary 38. The opposite extreme inequality is

w-sat(n, T) ≥ e(T) − 1, n ≥ m.  (75)

By Lemma 35, if we have equality in (75), then T admits a g-proof for n. In fact, we can show that we have a G-proof.

Lemma 55 Let F and H be any forests with e(F) ≤ e(H). Then F independent in G_{H}.

Proof. We use induction on l = e(H). It is enough to prove the claim for e(F) = e(H). Assume that 1 is an endvertex incident to the edge E = {1, 2} in both F and H. Clearly,

det(M(H, F)) = ±α_{1,1}α_{2,2} det(M(H − E, F − E)) + (α_{1,1}-free polynomial).

By induction we conclude that det(M(H, F)) ≠ 0, which proves the lemma.

Corollary 56 If G ∈ w-SAT(n, T), for some forest T, and e(G) = e(T) − 1 then the pair (T, G) admits a G-proof.

Proof. Indeed, G is a forest. Also, T is dependent in G_{G} but, by Lemma 55, any proper subgraph of T is not; hence T is a G_{G}-circuit.

If T contains, for example, vertices a, b, c of degrees 1, 1, 2 respectively such that {a, c}, {c, d}, {b, d} ∈ E(T), for some vertex d, then adding the edges {d, x} and {x, y} to T, any x, y ∉ V (G), we create each time a new graph isomorphic to T; this implies equality in (75) with possible exceptions for some n ≤ 2m. Generating a random tree by, for example, taking all m^{m−2} vertex-labelled trees equiprobable, one can show that almost every tree contains the above ‘abcconfiguration’ and therefore admits a G-proof.

The above results can be extended to hypertrees, for the definitions see Part IV, but we do not want to clutter the text with details.

11 Cones

In this section we prove that cones ‘preserve’ G/g/g′-proofs. These results appear in [Pik99a].

To define the cone cn(G) of an r-graph G, add to G a new vertex and all edges containing this vertex. In other words, pick v ∉ V (G) and define V (cn(G)) = V (G) ∪ {v} and

E(cn(G)) = E(G) ∪ {{v} ∪ E : E ∈ V (G)^{(r−1)}} .

For a family F of r-graphs, define cn(F) = {cn(F) : F ∈ F}.

For 2-graphs, cn^{l}(F) = K_{l} + F; so, for example, the cones of empty graphs, cycles, complete graphs are stars, wheels and complete graphs respectively.

Lemma 57 Suppose that every r − 1 vertices of an r-graph F are covered by at least one edge. If F is a G_{G}-chain, for some r-graph G, then cn(F) is a G_{cn(G)}-chain.

Proof. Suppose first that v(G) ≥ v(F). Let G′ = cn(G), V (G) = [n − 1] and V (G′) = [n]. Identify the vertices of G′ with the basis {e_{1}, …, e_{n}} of a vector space V′. Let Z′ be the subspace of ⋀^{r}V′ and let G_{G′} be the gross matroid on [n]^{(r)} corresponding to G′.

We may assume that F′ = cn(F) is embedded into [n] so that V (F′)\V (F) = {n}. We have to show that E(F′) is a chain in G_{G′}, that is, we have to find h′ ∈ Z′ such that supp(h′) = E(F′). Define g_{n}^{∗} = f_{n}^{∗} and

g_{i}^{∗} = f_{i}^{∗} − \frac{α_{in}}{α_{nn}} f_{n}^{∗},  i = 1, …, n − 1.  (76)

Recall that f^{∗} is a generic basis of (V′)^{∗} and α_{ij} = f_{i}^{∗}(e_{j}), so g_{i}^{∗}(e_{n}) = 0, i ∈ [n − 1], and this is the main point of our definition.

The matrix N = (g_{i}^{∗}(e_{j}))_{i,j∈[n−1]} is a generic matrix for a generic choice of the α’s. Indeed, if its entries, γ_{ij} = α_{ij} − α_{in}α_{nj}/α_{nn}, i, j ∈ [n − 1], are algebraically dependent, then clearly the α’s are.

As F is a G_{G}-chain, the system of linear equations

g_{D}^{∗} ⌞ ( ∑_{E∈E(F)} c_{E}e_{E}) = 0,  D ∈ E(G),  (77)

with respect to the undeterminants (c_{E})_{E∈E(F)}, has a solution with all c’s being non-zero for generic g (which is the case for generic f). Apply elementary matrix transforms to write the system (77) in a diagonal form. For the free variables choose β_{1}, …, β_{k} which (together with the α’s) are algebraically independent over the rationals and compute the other variables each being a rational function of the α’s and β’s.

Let h = ∑_{E∈E(F)} c_{E}e_{E} and h′ = f_{n}^{∗} ⌞ (h ∧ e_{n}). To complete the theorem it is enough to show that h′ ∈ Z′ and supp(h′) = E(F′).

Let D ∈ E(G′). We want to show that f_{D}^{∗} ⌞ h′ = 0. If D ∋ n, then

⟨f_{D}^{∗}, h′⟩ = ⟨f_{D\{n}}^{∗} ∧ f_{n}^{∗}, f_{n}^{∗} ⌞ (h ∧ e_{n})⟩ = 0.

If n ∉ D, that is, D = {d_{1}, …, d_{r}} ∈ E(G), then, by (76),

f_{D}^{∗} = ⋀_{i=1}^{r}f_{d_{i}}^{∗} = ⋀_{i=1}^{r} (g_{d_{i}}^{∗} + \frac{α_{d_{i}n}}{α_{nn}} f_{n}^{∗}) = g_{D}^{∗} + f_{n}^{∗} ∧ x^{∗},

some x^{∗} ∈ ⋀^{r−1}V^{∗}. Now,

⟨f_{D}^{∗}, h′⟩ = ⟨g_{D}^{∗} + f_{n}^{∗} ∧ x^{∗}, f_{n}^{∗} ⌞ (h ∧ e_{n})⟩ = ⟨g_{D}^{∗} ∧ f_{n}^{∗}, h ∧ e_{n}⟩.

But for every i ∈ [n − 1] we have g_{i}^{∗}(e_{n}) = 0, so the above expression is equal to f_{n}^{∗}(e_{n})⟨g_{D}^{∗}, h⟩ which is zero by the definition of h. Therefore h′ ∈ Z′.

Let us show that supp(h′) = E(F′). Clearly, every E ∈ supp(h′) either contains n or belongs to E(F) which shows that supp(h′) ⊂ E(F′). On the other hand, take any E ∈ E(F′). If E ∈ E(F), then

⟨e_{E}^{∗}, h′⟩ = ⟨e_{E}^{∗} ∧ f_{n}^{∗}, h ∧ e_{n}⟩ = ⟨e_{E}^{∗}, h⟩ · ⟨f_{n}^{∗}, e_{n}⟩ = c_{E}f_{n}^{∗}(e_{n}) ≠ 0,

because n ∉ E. If E ∋ n, then let D_{1}, …, D_{l} be the edges of F containing E′ = E \ {n}. By our assumption, l > 0. Let D_{i} \ E = {d_{i}}. Then

P_{E} = ⟨e_{E}^{∗}, h′⟩ = ⟨e_{E′}^{∗} ∧ e_{n}^{∗} ∧ f_{n}^{∗}, h ∧ e_{n}⟩ = −⟨e_{E′}^{∗} ∧ f_{n}^{∗}, h⟩
= −⟨e_{E′}^{∗} ∧ f_{n}^{∗}, ∑_{E∈E(F)} c_{E}e_{E}⟩ = ∑_{i=1}^{l} ±c_{D_{i}}⟨f_{n}^{∗}, e_{d_{i}}⟩ = ∑_{i=1}^{l} ±c_{D_{i}}α_{n,d_{i}}.

(The third equality is true as supp(h) = E(F) ⊂ [n − 1]^{(r)}.)

As every c_{D_{i}} is a rational function in the α’s and β’s, so is P_{E}. To show that P_{E} ≠ 0 for a generic f, it is enough to demonstrate an example of f when P_{E} ≠ 0. Let α_{in} = 0, i ∈ [n − 1]. Then system (77) reduces to

f_{D}^{∗} ⌞ ( ∑_{E∈E(F)} c_{E}e_{E}) = 0,  D ∈ E(G).  (78)

By the algebraic independence of (f_{i}^{∗}(e_{j}))_{i,j∈[n−1]}, if we perform the diagonalisation for (78) in the same order as for (77), we will obtain the same set of free variables. Therefore, (c_{E})_{E∈E(F)} provides every solution for (78) when the β’s range over the reals. Thus each c_{E} is non-zero (as F is a G_{G}-chain) and it can depend only on f_{i}^{∗}(e_{j}) = α_{ij}, i, j ∈ [n − 1], and the β’s. Now it is obvious that P_{E} = ∑_{i=1}^{l} c_{D_{i}}α_{n,d_{i}} cannot be identically zero. This proves the lemma if v(G) ≥ v(F).

Otherwise, we can add v(F) − v(G) isolated vertices to G to obtain H. By above, cn(F) is a chain in G_{cn(H)}, that is, each edge of cn(F) is dependent on the other edges. The latter claim is certainly true in G_{cn(G)} which has more dependences than G_{cn(H)} as cn(G) ⊂ cn(H).

Lemma 58 If an r-graph F is independent in G_{G} and v(F) ≤ v(G), then cn(F) is independent in G_{cn(G)}.

Proof. We assume the same conventions as those appearing in the proof of Lemma 57 before (77).

It is enough to prove our claim in the case e(G) = e(F): if e(G) > e(F) we can remove a G-edge with F being still G_{G}-independent.

Let us show that the rank of M′(G′, F′) is e(F′), where M′(D, E) = ⟨g_{D}^{∗}, e_{E}⟩, D ∈ E(G′), E ∈ E(F′), which would imply the lemma.

By our assumption, the square submatrix M′(G, F) ⊂ M′(G′, F′) is nonsingular because the matrix N is generic. As g_{i}^{∗}(e_{n}) = 0 for i ∈ [n − 1], we conclude that all entries of the submatrix M′(G, F′′) are zeros, where E(F′′) = E(F′)\E(F). Therefore, to prove the claim we have to show that the submatrix M′(G′′, F′′) has the maximal possible rank \binom{v(F)}{r−1}, where E(G′′) = E(G′)\E(G).

For any D′ = D ∪ {n} ∈ E(G′′), E′ = E ∪ {n} ∈ E(F′′), we have

⟨g_{D′}^{∗}, e_{E′}⟩ = g_{n}^{∗}(e_{n}) · ⟨g_{D}^{∗}, e_{E}⟩,

because g_{i}^{∗}(e_{n}) = 0, i ∈ [n − 1]. (As n is the last element in D′ and E′, we do not have ±1 in the formula.) Now,

M′(G′′, F′′) = g_{n}^{∗}(e_{n}) · M′(K^{r−1}([n − 1]), K^{r−1}(V (F)))

has rank \binom{v(F)}{r−1} because N is generic.

Remark. Is is not hard to show that if F is not independent in G_{G}, then cn(F) is not independent in G_{cn(G)}, for any r-graphs F and G. But we do not need this result.

Lemma 59 If G ∈ w-SAT(n − 1, F), then cn(G) ∈ w-SAT(n, cn(F)). In particular,

w-sat(n, cn(F)) ≤ w-sat(n − 1, F) + \binom{n − 1}{r − 1}.

Proof. Let E_{1}, …, E_{m} be an F-proper ordering of E(\overline{G}). To show that G′ = cn(G) is weakly cn(F)-saturated, add these edges in the same order to G′. (Note that E(\overline{G′}) = E(\overline{G}).) Every E_{i} creates an F -subgraph in G, F ∈ F, which, together with the extra vertex, creates a copy of cn(F) in G′, so G′ ∈ w-SAT(n, cn(F)).

Theorem 60 Let F be a family of r-graphs such that in each F ∈ F every r−1 vertices are covered by at least one edge.

If a pair (F, G) admits a G-proof, then the pair (cn(F), cn(G)) admits a G-proof.

If we can g-prove w-sat(n − 1, F) ≥ l, then we can g-prove

w-sat(n, cn(F)) ≥ l + \binom{n − 1}{r − 1}.  (79)

In particular, if F admits a g-proof for n − 1, then cn(F) admits a g-proof for n. The analogous claim is true for the g′-technique.

Proof. Let us consider G-proofs first. By Lemma 59, cn(G) is weakly cn(F)- saturated. By Lemma 57, cn(F) is a G_{cn(G)}-chain for every F ∈ F. Hence, the pair (cn(F), cn(G)) admits a G-proof.

Next, consider the g-technique. Take any G such that each F ∈ F is a G_{G}-chain and R_{G_{G}}(K_{n−1}^{r}) ≥ l. Adding extra vertices to G, we may assume v(G) ≥ n − 1. By Lemma 57, each graph in cn(F) is a chain in G_{cn(G)}.

By definition, R_{G_{G}}(K_{n−1}^{r}) ≥ l, so choose a G_{G}-independent subgraph H ⊂ K_{n−1}^{r} of rank l. Assume v(H) = n − 1. By Lemma 58, cn(H) is independent in G_{cn(G)}. Hence, the rank of K_{n}^{r} in G_{cn(G)} is at least e(cn(H)) = l + \binom{n−1}{r−1}, that is, we can g-prove (79), as required.

In the g′-case, choose G such that each F ∈ F is a G_{G}-chain and

R_{G_{G}}(K_{n−1}^{r}) + D_{G_{G}}(F) − 1 ≥ l.

Now we proceed in the same way as in the g-case, except we have to show additionally that, for any F ∈ F, we have D_{G_{G}}(F) ≤ D_{G_{cn(G)}}(cn(F)).

Note that if we have F -edges E_{1}, …, E_{d} whose removal does not decrease the G_{G}-rank of E(F), then the system of equations (77) has a solution in which c_{E_{1}}, …, c_{E_{d}} can be chosen to be the free variables β_{1}, …, β_{d}. Following the proof of Lemma 57 (note that F is a G_{G}-chain), one can let (c_{E}) be such a solution of (77) and observe that

⟨e_{E_{i}}^{∗}, h′⟩ = ⟨e_{E_{i}}^{∗} ∧ f_{n}^{∗}, h ∧ e_{n}⟩ = ⟨e_{E_{i}}^{∗}, h⟩ · ⟨f_{n}^{∗}, e_{n}⟩ = β_{i}α_{nn}, i ∈ [d],

since E_{i} ⊂ [n − 1]. This means that, choosing generic β’s, we can obtain h′ ∈ Z′ whose support is E(cn(F)) with e_{E_{i}}^{∗}(h′) being generic, which is precisely to say that E_{1}, …, E_{d} are G_{cn(G)}-dependent on the other edges of cn(F). Hence, D_{G_{cn(G)}}(cn(F)) ≥ d and the claim follows.

Remark. We cannot generally discard the covering condition in Lemma 57 or Theorem 60. (But note that we do not have any covering condition on G.) Consider, for example, r = 2 when the condition rules out isolated vertices. Let F be a triangle plus an isolated vertex and let G be a star K_{1,n−2}, n ≥ 5. Then (F, G) admits a G-proof (see Subsection 10.1). But it is easy to see that w-sat(n, cn(F)) = 6 < e(cn(G)) = 2n − 3, and so cn(F) cannot be a G_{cn(G)}- chain.

We noted already in Section 7 that many new results can be proved by applying Theorem 60, so we do not repeat these examples here.

12 Joins

Here we indicate how to extend the idea of G/g/etc.-proof to layered graphs (which were defined in Subsection 4.2) and prove that joins ‘preserve’ G/g/rproofs. These results appear in [Pik99a].

The notion of weak saturation extends to layered graphs in the obvious way. For example, given an r-graph F, w-SAT(n, F) consists of all r-graphs G on an n-set such that we can consecutively add all missing r-edges to G creating every time an F-subgraph.

It is clear how to extend the notion of an m/r-proof to layered graphs. It is possible also to introduce the gross matroid of an r-graph G defined on an n-set X. Indeed, identify each X_{i} with a basis e_{i} = (e_{i,j})_{j∈[n_{i}]} of an n_{i}-dimensional vector space V_{i} and consider ⋀V which, by the definition, is the tensor product of the exterior algebras over V_{i}, i ∈ [t]:

⋀V = ⨂_{i∈[t]} ⋀V_{i}.

Let ⋀^{r}V be the linear subspace of ⋀V spanned by the elements

h = h_{1} ⊗ … ⊗ h_{t}, h_{i} ∈ ⋀^{r_{i}}V_{i}, i ∈ [t].

Let f_{i} = (f_{i,j})_{j∈[n_{i}]} be another basis of V_{i} lying in generic position with respect to e_{i}, i ∈ [t].

In the obvious way we define supports, etc. For any r-subset E ⊂ X, let

f_{E} = ⨂_{i∈[t]} f_{i,E_{i}} and e_{E} = ⨂_{i∈[t]} e_{i,E_{i}},

or, in other words, in every ⋀V_{i}, we take the element corresponding to E_{i} in the basis f_{i} or e_{i} and then compute the tensor product. Let the linear subspace Z ⊂ ⋀^{r}V corresponding to G be spanned by the elements \{f_{E} : E ∈ E(\overline{G})\} and let rsets E_{1}, …, E_{k} be independent if no linear combination of e_{E_{1}}, …, e_{E_{k}} (except

0) belongs to Z. The required matroid G_{G} of rank codim(Z) = e(G) is built. Clearly, it is symmetric, that is, invariant under layer-preserving permutations.

Given t (usual) r_{i}-graphs F_{i}, i ∈ [t], with disjoint vertex sets, their join (or tensor product) F = F_{1} ⊗ … ⊗ F_{t} is the layered r-graph on the layered set V (F) = (V (F_{1}), …, V (F_{t})) such that an r-subset E = (E_{1}, …, E_{t}) is an edge of F if and only if E_{i} ∈ E(F_{i}) for every i ∈ [t]. Thus e(F) = ∏_{i∈[t]} e(F_{i}). For example, the join of two 1-graphs is a complete bipartite graph (possibly plus isolated vertices).

Suppose that we are given t families F_{i} of r_{i}-graphs, i ∈ [t]. We define their join by

F = F_{1} ⊗ … ⊗ F_{t} = {F_{1} ⊗ … ⊗ F_{t} : F_{i} ∈ F_{i}, i ∈ [t]} .

Let these conventions apply to the following results.

Lemma 61 If G_{i} ∈ w-SAT(n_{i}, F_{i}), i ∈ [t], then G ∈ w-SAT(n, F), where \overline{G} = \overline{G_{1}} ⊗ … ⊗ \overline{G_{t}}. In particular,

w-sat(n, F) ≤ ∏_{i∈[t]} \binom{n_{i}}{r_{i}} − ∏_{i∈[t]} (\binom{n_{i}}{r_{i}} − w-sat(n, F_{i})) .

Proof. Denote b_{i} = e(\overline{G_{i}}). Let E_{i,j} ∈ \overline{G_{i}}, j = 1, …, b_{i}, be an F_{i}-proper ordering, i ∈ [t]. There is the obvious bijective correspondence between the elements in B = [b_{1}] × … × [b_{t}] and the edges of \overline{G} which maps (j_{1}, …, j_{t}) to ∪_{i∈[t]}E_{i,j_{i}}.

Now we add the missing edges to G so that the corresponding elements of B are taken in the lexicographic order. Consider any added edge E. Let H_{i} ⊂ X_{i} be an F_{i}-subgraph created by E_{i}. (Note that E_{i} ∉ E(G_{i}) by the definition of G.) We claim that H = H_{1} ⊗ … ⊗ H_{t} is a forbidden subgraph created by E. Indeed, let D ≠ E, be an edge of H. Clearly, for each i ∈ [t], the edge D_{i} ∈ E(H_{i}) must be present in G_{i} or be added before E_{i} or equal to E_{i}. If D_{i} ∈ E(G_{i}) for at least one index i then D ∈ E(G). If not, then clearly the edge D comes before E, as required.

Finally, e(\overline{G}) = ∏_{i∈[t]} e(\overline{G_{i}}), which completes the proof.

Lemma 62 If F_{i} is a chain in G_{G_{i}}, i ∈ [t], then F = F_{1} ⊗ … ⊗ F_{t} is a chain in G_{G}, where \overline{G} = \overline{G_{1}} ⊗ … ⊗ \overline{G_{t}}.

Proof. By the assumption, there is h_{i} ∈ Z_{G_{i}} ⊂ ⋀^{r_{i}}V_{i} such that supp_{e_{i}}(h_{i}) = E(F_{i}), i ∈ [t]. Consider

h = h_{1} ⊗ … ⊗ h_{t} ∈ ⋀^{r}V.

Obviously, supp_{e}(h) = E(F) and supp_{f} (h) ⊂ E(\overline{G}). Therefore, h ∈ Z_{G} and every edge in F is dependent on the rest, as required.

Theorem 63 Suppose that, for every i ∈ [t], the pair (F_{i}, G_{i}) admit a G-proof. Then so does the pair (F, G), where \overline{G} = \overline{G_{1}} ⊗ … ⊗ \overline{G_{t}}.

Suppose that, for each i ∈ [t], we can g-prove that w-sat(n_{i}, F_{i}) ≥ l_{i}. Then we can g-prove that

w-sat(n, F) ≥ ∏_{i∈[t]} \binom{n_{i}}{r_{i}} − ∏_{i∈[t]} (\binom{n_{i}}{r_{i}} − l_{i}) .  (80)

In particular, if each F_{i} admits a g-proof for n_{i}, then F admits a g-proof for n. The analogous statement is true for the r-technique.

Proof. Let us consider G-proofs first. By Lemma 61, G ∈ w-SAT(n, F), and by Lemma 62, every F_{1} ⊗ … ⊗ F_{k} ∈ F is a G_{G}-chain, and the claim follows.

Now, consider the g-case. For i ∈ [t], choose G_{i} such that each graph in F_{i} is a G_{G_{i}}-chain and the G_{G_{i}}-rank of K_{n_{i}}^{r_{i}} is at least l_{i}; let H_{i} ⊂ K_{n_{i}}^{r_{i}} be a G_{G_{i}}-independent subgraph of size l_{i} and order n_{i}. Let

\overline{G} = \overline{G_{1}} ⊗ … ⊗ \overline{G_{t}},
\overline{H} = \overline{H_{1}} ⊗ … ⊗ \overline{H_{t}}.

By Lemma 62, each F_{1} ⊗ … ⊗ F_{k} ∈ F is a G_{G}-chain.

Let us show that H is independent in G_{G}. As each H_{i} is G_{G_{i}}-independent, we can find a linear map p_{i} : ⋀^{r_{i}}V_{i} → Z_{G_{i}} which is the identity map on Z_{G_{i}} while p_{i}(e_{E}) = 0 if E ∈ E(H_{i}), i ∈ [t]. Define

p = p_{1} ⊗ … ⊗ p_{t} : ⋀^{r}V → Z_{G_{1}} ⊗ … ⊗ Z_{G_{t}},

that is p(u_{1} ⊗ … ⊗ u_{t}) = p_{1}(u_{1}) ⊗ … ⊗ p_{t}(u_{t}). Now, p is the identity map on Z_{G_{1}} ⊗ … ⊗ Z_{G_{t}} = Z_{G}, while p is zero on e_{E} for each E = E_{1} ∪ … ∪ E_{t} ∈ E(H): E_{i} ∈ E(H_{i}) for some i ∈ [t] and then p_{i}(e_{E_{i}}) = 0. Hence, no non-zero linear combination of e_{E}, E ∈ E(H) can lie in Z_{G}, that is, H is independent in G_{G}. The size of H equals the right-hand side of (80), as required.

The claim about g-proofs follows from Lemma 61.

In the r-case, our task is to construct a matroid M on the set of r-subsets of X such that every graph in F is an M-chain, should we be given appropriate matroids M_{i} on Y_{i} = X_{i}^{(r_{i})}, i ∈ [t].

Let k_{i} : Y_{i} → V_{i}, for some vector space V_{i}, be a representation of the matroid M_{i}, i ∈ [t]. Identify Y_{i} with a basis of some vector space W_{i} via g_{i} : Y_{i} ↪ W_{i}. Let h_{i} : W_{i} → V_{i} be the linear map extending k_{i}. Denote Z_{i} = ker(h_{i}) ⊂ W_{i}. Clearly, codimZ_{i} = R_{M_{i}}(Y_{i}) = e(G_{i}) ≥ l_{i}, where G_{i} is a base of M_{i}.

Let \overline{G} = \overline{G_{1}} ⊗ … ⊗ \overline{G_{t}}. Identify the r-subsets of V (G) with a basis of W = ⨂_{i∈[t]} W_{i} by mapping E = (E_{1}, …, E_{t}) into g(E) = ⨂_{i∈[t]} g_{i}(E_{i}). Let Z = ⨂_{i∈[t]} Z_{i} ⊂ W and p : W → W/Z be the projection.

Let M be the matroid represented by p ∘ g : V (G)^{(r)} → W/Z. Let us show that M r-proves (80).

As g(V (G)^{(r)}) is a basis for W, we conclude that the rank of M is

dim W − dim Z = ∏_{i∈[t]} \binom{n_{i}}{r_{i}} − ∏_{i∈[t]} e(\overline{G_{i}}),

which is at least the right-hand side of (80).

Thus, all we have to do is to check that any F = F_{1} ⊗ … ⊗ F_{t} ∈ F is an M-chain. Fix an edge E = (E_{1}, …, E_{t}) ∈ E(F). As F_{i} is an M_{i}-chain, we conclude that there are c_{i,E} ∈ ℝ, E ∈ E(F_{i}) \ {E_{i}}, and z_{i} ∈ Z_{i} such that

g_{i}(E_{i}) = z_{i} + ∑_{D∈E(F_{i})\{E_{i}}} c_{i,D}g_{i}(D),  i ∈ [t].  (81)

If we take the tensor product of (81) over i ∈ [t], we obtain on the left-hand side the element g(E) while on the right-hand side we will have z_{1} ⊗ … ⊗ z_{t} ∈ Z plus some other tensor products. Next, in the remaining tensor products replace each z_{i} by the linear combination of (g_{i}(D))_{D∈E(F_{i})} by (81). Each term then becomes ⊗_{i∈[t]}g_{i}(D_{i}) for some D_{i} ∈ E(F_{i}), i.e., it is of the form g(D), D = (D_{1}, …, D_{t}) ∈ E(F) and, moreover, we never have D = E. So we have a representation of g(E) as a linear combination of an element of Z and of g(D), D ∈ E(F) \ {E} which is precisely the required. The theorem is proved.

Unfortunately, there does not seem to be a natural tensor product operation for matroids, cf. Lovász [Lov77], so we do not know if joins preserve m-proofs.

Alon [Alo85] (a different proof is presented by Yu [Yu93]) solved one extremal problem for set systems, which can be easily seen equivalent to computing the w-sat-function for joins of complete graphs. As complete graphs admit a Gproof (e.g. by Theorem 60), the result of Alon can be deduced as a special case of Theorem 63.

Part III

Chain Decompositions

13 Introduction

13.1 Discussion

There are many important results about chain decompositions of posets, that is, collections of chains such that every element in the poset belongs to exactly one chain. (We will also refer to these as vertex decompositions.) Typical questions are the following. What is the minimal number of chains of such a partition? Do there exist partitions with some extra properties (e.g. into symmetric chains)? Are there any applications of these decompositions?

In this part we investigate the notion of an edge decomposition which is a collection of chains such that every pair of adjacent elements (one covers the other) belongs to exactly one chain and we try to answer the above questions.

Such considerations may arise, for example, when in a computer programme we want to operate with posets, and so we wish to represent them efficiently in the memory. If keeping the relational binary n × n-table is impossible or undesirable, we can try to maintain a list of chains completely determining the poset, and a natural question to ask is, for example, how small such a list can be. The related notion of line poset also arises naturally.

In Section 14 we compute the minimal size of a skipless chain decomposition of a poset in terms of other parameters, which can be viewed as an analogue of Dilworth’s theorem [Dil50]. Surprisingly, this fundamental theorem is a new result. We prove it using the linear programming method of Dantzig and Hoffman [DH56]. Graham Brightwell simplified our proof by replacing the linear programming argument by an easy application of Hall’s theorem. We present both these proofs.

The minimal size of an edge decomposition of P can be deduced as a corollary but we present a short and direct proof.

In Section 15 we provide an explicit edge decomposition of the lattice of subsets of a finite set into symmetric chains. Although the existence of such a partition can be deduced from the results of Anderson [And67] and Griggs [Gri77], a constructive proof seems to be unknown. The discovered partition has some extra properties and interesting applications. For the latter we refer the reader to Section 16.

In Section 17 we characterize line posets in terms of forbidden configurations and point out which information determines and can be reconstructed from its line poset.

13.2 Definitions

Let P = (X, >) be a poset (a partially ordered set). We say that y covers x (denoted by y ⋗ x or x ⋖ y) if y > x and no z ∈ X satisfies x < z < y (such x, y will be also called adjacent elements). With every poset P we associate its Hasse diagram D = D(P) which is the digraph with X as the vertex set and (x, y) ∈ E(D) iff y covers x. Given a cycle-free digraph D, we can build a poset on the same vertex set by letting x < y if there is a directed xy-path. Note that a cycle-free digraph D is the Hasse diagram of some P if and only if for every (x, y) ∈ E(D) there is no directed xy-path of length greater than 1. The correspondence ‘posets-digraphs’ is very useful, so we often switch between the poset and digraph terminology without any warning.

A chain in P is called skipless if every element covers its predecessor; skipless chains correspond to oriented paths in the Hasse diagram. The width w(P) is the maximal size of an antichain in P.

The line poset L(P) of a poset P has as the vertex set the pairs (x, y) of elements in P with y covering x and we agree that (x ⋖ y) is less than (x′ ⋖ y′) in L(P) if and only if y ≤ x′. (This operation somewhat resembles taking the line graph, hence the name.)

Every skipless chain in P corresponds to a skipless chain in L(P) of size smaller by 1. We usually identify these chains.

One can ask which important poset properties are preserved by the operator L. In fact, L preserves very few properties (e.g. self-duality, regularity). As in almost every case it is trivial to find a counterexample/proof, we do not dwell on this topic.

A vertex partition (decomposition) of P is a collection of chains such that every x ∈ X belongs to exactly one chain. An edge partition (decomposition) is a family of skipless chains such that every pair x, y ∈ X with x being covered by y belongs to exactly one chain. Note that the chains in an edge decomposition are required to be skipless. One can see that edge partitions of P correspond to vertex partitions of L(P) into skipless chains.

The subsets of [n] partially ordered via the inclusion relation, form the ranked poset B_{n} = (2^{[n]}, ⊂). The corresponding Hasse diagram is the oriented n-cube Q_{n}. For B_{n}, the relation ‘B covers A’ is denoted by A ⊏ B.

We find it useful to identify A ∈ B_{n} with its ()-representation which is the n-sequence of left and right parentheses corresponding to the elements of \overline{A} = [n] \ A and A respectively. Likewise, the (∗)-representation of an element (A ⊏ B) ∈ L(B_{n}) contains ‘(’ for the elements in \overline{B}, ‘)’ for the elements in A and ‘∗’ for the element in B \ A.

Generally, let F be a sequence containing left and right parentheses. Consecutively and as long as possible remove matched pairs of adjacent brackets, ie. substrings ‘( )’. (Clearly, the order of operations does not matter.) The elements which would be removed by the above matching are called fixed or paired elements and the remaining ones are called free. In particular, the free parentheses always form the following (possibly empty) sequence: ) ) …) ) ( ( … ( (.

14 Skipless Chain Decompositions

Here we present a theorem computing the minimal number of skipless chains partitioning a given poset P. In fact, we prove a more general result about directed graphs.

Let D be any digraph. We may have loops and may have edges (i, j) and (j, i) simultaneously. Consider partitions of V (D) into vertex-disjoint directed cycles and directed paths. (We consider any isolated vertex as a path of length zero; loops and pairs of opposite edges are considered as cycles.) Let m(D) be the minimal number of directed paths in a such partition.

On the other hand, let M(D) be the maximal value of |A| − |B| taken over all pairs of disjoint sets A, B ⊂ V (D) such that any directed path connecting two distinct vertices from A, contains a vertex of B and any cycle intersecting A intersects B. (In particular, if (i, i) ∈ E(D) then i ∉ A.) Clearly, for any such pair (A, B) we have |P ∩ A| ≤ |P ∩ B| + ε, where ε = 1 if P is a directed path and ε = 0 if P is a directed cycle. This implies that m(D) ≥ M(D).

We will show that we have in fact equality for any D. Our proof is a modification of the proof by Dantzig and Hoffman [DH56] of Dilworth’s theorem, which exploits methods of linear programming. (A simpler argument by Graham Brightwell is outlined after our proof.)

Theorem 64 For any directed graph D we have m(D) = M(D).

Proof. As we have already observed m(D) ≥ M(D), so let us prove the converse inequality. Assume that V (D) = [n]. For i, j ∈ [n] define c_{00} = 1, c_{0j} = c_{i0} = 0, and

c_{ij} = \{ \begin{matrix}0, & if (i, j) ∈ E(D), \\ −∞, & if (i, j) ∉ E(D). \end{matrix}

Consider the linear programming problem of finding k, where

k = max ∑_{i,j∈[0,n]} c_{ij}x_{ij},  (82)

given the following restrictions:

∑_{j=0}^{n} x_{0j} = ∑_{i=0}^{n} x_{i0} = n,  (83)
∑_{j=0}^{n} x_{ij} = ∑_{j=0}^{n} x_{ji} = 1, i ∈ [n],  (84)
x_{ij} ≥ 0, i, j ∈ [0, n].  (85)

Restrictions (83), (84) and (85) define a non-empty set; for example, we can satisfy them by letting x_{ij} be 0 for i, j ∈ [n] and 1 otherwise, except x_{00} = 0. As for any feasible solution we have x_{00} ≤ n while the coefficients c_{ij} at other variables are non-positive, we conclude that the right-hand side of (82) is at most n and thus k is well-defined.

We claim that we can choose an integral solution to (82), that is, we can ensure that each x_{ij} is an integer. To do so, take a solution in which as many as possible variables are integers. Suppose there is x_{i_{1}i_{2}} ∉ ℤ. By (83) or (84), the i_{2}th column contains another non-integer, x_{i_{3}i_{2}}. Next, we consider the i_{3}th row and find x_{i_{3}i_{4}} ∉ ℤ, and so on, until considering a current variable x_{i_{s}i_{t}} we have a chance to select a previously chosen variable x_{i_{u}i_{v}}. In fact, we may have two choices at this step, but we will always be able to chose one with s + v and t + u being odd. Then the subsequence S of elements between x_{i_{s}i_{t}} and x_{i_{u}i_{v}} (inclusive) is of even length. If we add any ε to each x_{i_{k}i_{k+1}} ∈ S and subtract ε from each x_{i_{k+1}i_{k}} ∈ S, then we do not affect (83) and (84). (Because each row or column contains either two variables, which receive different signs, or none.) The function ∑ c_{ij}x_{ij} is linear in ε, suppose it is non-decreasing. Let ε be the minimum of the fractional part of x_{i_{k+1}i_{k}} ∈ S; then our transformation makes at least one more variable integral, while (85) still holds. This contradiction proves the claim.

Any x_{ij}, except perhaps x_{00}, is either 0 or 1. A moment’s thought reveals that by (84) the set {(i, j) ∈ [n]^{2} : x_{ij} = 1} ⊂ E(D) is a union of vertex-disjoint directed paths (this is to include isolated vertices) and cycles partitioning V (D). The number of paths equals the number of occurrences of 1 among x_{0j}, j ∈ [n], which by (83) is n − x_{00} = n − k. Hence,

m(D) ≤ n − k.  (86)

Now, the Duality Theorem asserts that

k = min (n(u_{0} + v_{0}) + ∑_{i=1}^{n} u_{i} + ∑_{j=1}^{n} v_{j}) ,  (87)

given the following restrictions on variables u_{i}, v_{i}, i ∈ [0, n]:

u_{0} + v_{0} ≥ 1,  (88)
u_{i} + v_{0} ≥ 0, i ∈ [n],  (89)
u_{0} + v_{j} ≥ 0, j ∈ [n],  (90)
u_{i} + v_{j} ≥ 0, (i, j) ∈ E(D).  (91)

We claim that we can choose an integral solution to (87). To do so, take a solution with as many as possible variables among u_{i}, v_{i}, i ∈ [0, n], being integers. Let I = {i ∈ [0, n] : u_{i} ∉ ℤ} and J = {j ∈ [0, n] : v_{j} ∉ ℤ}. Suppose I ≠ ∅. If we decrease each u_{i}, i ∈ I, by ε and increase each v_{j}, j ∈ J, by ε, then the right-hand side of (87) is linear in ε; suppose it non-increases with ε. Let ε = min_{i∈I}(u_{i} − ⌊u_{i}⌋). We obtain at least one more integer among the u’s, so to obtain a contradiction it is enough to check that (88)–(91) still hold. The restriction (88), for example, may cause us a problem only if 0 ∈ I \ J. Then v_{0} ∈ ℤ and, by definition, ε is at most the fractional part of u_{0} +v_{0} which would suffer the decrement by ε without crossing the integer 1. The claim is proved.

We may assume that v_{0} = 0, because we can add some integer ε to each u_{i} and subtract ε from each v_{i}, i ∈ [0, n], without affecting (87)–(91). Also, we can make u_{0} = 1, because we can subtract ε > 0 from u_{0} and add ε to each v_{i}, i ∈ [n]. Hence, k = n + min(∑_{i=1}^{n} u_{i} + ∑_{j=1}^{n} v_{j}), given conditions u_{i} ≥ 0, v_{i} ≥ −1, i ∈ [n], and (91). It is easy to see that in our (integral) solution, each u_{i} is either 0 or 1 and each v_{i} is either −1 or 0. Let X = {j ∈ [n] : v_{j} = −1} and let X′ = {i ∈ [n] : ∃j ∈ X (i, j) ∈ E(D)}, that is, X′ consists of vertices sending at least one edge to X.

To satisfy (91) we must have u_{i} = 1 for each i ∈ X′. Also, if we set u_{i} = 0 for i ∈ [n] \ X′, then (88)–(91) are still satisfied while the linear function in (87) does not increase. Hence, we may assume that X′ = {i ∈ [n] : u_{i} = 1}; then n − k = |X| − |X′|.

Let A = X \ X′ and B = X′ \ X. Let P = {x_{1}, …, x_{l}} be a directed path in D with x_{1}, x_{l} ∈ A, l ≥ 2. As x_{1} ∉ X′, we conclude x_{2} ∉ X. As x_{l} ∈ X, there must be i ∈ [2, l − 1] such that x_{i} ∉ X but x_{i+1} ∈ X. By definition, x_{i} ∈ B. Similarly, any cycle intersecting A intersects B. By (86) we obtain,

m(D) ≤ n − k = |X| − |X′| = |A| − |B| ≤ M(D),

which was required.

Remark. Graham Brightwell considerably simplified our proof shortly after it had been announced. Let us outline his argument which exploits Hall’s theorem.

Given a digraph D, consider the bipartite graph G on two copies of V (D), say X = {v^{∨} : v ∈ V (D)} and Y = {v^{∧} : v ∈ V (D)}, where we connect u^{∨} to v^{∧} if and only if (u, v) ∈ E(D). It is easy to check that the number of edges missing in a maximum matching in G equals m(D). By a version of Hall’s theorem, this number equals the maximum of |Z| − |Γ(Z)| over Z ⊂ X. Choose any extremal set Z. Now, it is routine to check that

A = {v ∈ V (D) : v^{∨} ∈ Z, v^{∧} ∉ Γ(Z)},
B = {v ∈ V (D) : v^{∨} ∉ Z, v^{∧} ∈ Γ(Z)},

are two sets exhibiting m(D) = |A| − |B| ≤ M(D).

The following corollary is obtained by applying Theorem 64 to the Hasse diagram of P.

Corollary 65 For any poset P, the minimal number m of skipless chains partitioning it equals the maximal value of |A| − |B| over all disjoint sets A, B ⊂ P such that any skipless chain containing two elements from A intersects B.

Of course, the minimal size m of an edge decomposition of P can be computed by applying Corollary 65 to L(P). However, we present a direct proof which is short and gives a direct algorithm for constructing such a partition. It turns out that to compute M(L(P)) it is enough to consider only pairs A, B ⊂ L(P) of the following rather special form: take a partition X ∪ Y = P and let A = {(x ⋖ y) ∈ L(P) : x ∈ X, y ∈ Y } and B = {(y ⋖ x) ∈ L(P) : x ∈ X, y ∈ Y }.

We state the result in terms of digraphs. Let e(X, Y), X, Y ⊂ V (D), denote the number of the edges in D starting in X and ending in Y and M(X, Y) = e(X, Y) − e(Y, X).

Theorem 66 The minimal number m of directed paths partitioning the edge set of a cycle-free digraph D is equal to

M(D) = max{M(X, Y) : X ∪ Y = V (D), X ∩ Y = ∅}.

Proof. It is immediate that m ≥ M because for any partition X ∪ Y = V (D) and any path P the removal of the edges on P can decrease M(X, Y) by at most one. To prove the reverse inequality by induction on |E(D)| it is enough to show that, for the graph D′ obtained from D by the removal of the edges of a maximal path P = (x_{1}, …, x_{k}), we have M(D′) < M(D).

To show this take a partition X ∪ Y = V (D′) with

M(D′) = M(X, Y) = e(X, Y) − e(Y, X).

Since P is maximal and D is acyclic there is no y ∈ V (D) with (y, x_{1}) ∈ E(D). Therefore if x_{1} ∈ Y we can move x_{1} to X without decreasing M(X, Y). Likewise we may assume x_{k} ∈ Y. But if we add back the edges of P we will increase M(X, Y) by 1: if moving along P we change side from Y to X i times, then we go from X to Y i + 1 times. This shows that M(D′) < M(D) as required.

Remark. Incidentally, we discovered an algorithm producing an optimal edge decomposition: select and remove maximal paths one by one.

15 Symmetric Edge Partitions of Cubes

The result of de Brujin, Tengbergen and Kruyswijk [BTK51] (see [Bol86, Theorem 4.1] or [And87, Section 3.1] for a proof) asserts that B_{n} = (2^{[n]}, ⊂) is a symmetric chain order, that is, admits a decomposition into symmetric chains. (A chain x_{1} < … < x_{k} in a ranked poset (P, r) is called symmetric if it is skipless and r(x_{1}) = r(P)−r(x_{k}).) This was strengthened by Anderson [And67] and Griggs [Gri77], who showed that a LYM poset P with a unimodal symmetric rank-sequence is a symmetric chain order. (Note that the number of chains is w(P)—minimal possible.)

The latter result is applicable to L(B_{n}) which, as a regular poset, has the LYM property, see e.g. [Eng97, Corollary 4.5.2]. However, this way we obtain a purely existential result while one would wish to have an explicit decomposition. Here we provide an explicit construction, which like that of Leeb (unpublished) and Greene and Kleitman [GK76] on B_{n}, utilizes bracket representations.

Theorem 67 L(B_{n}) is a symmetric chain order. In other words, B_{n} admits an edge decomposition into symmetric chains.

Proof. Assume that the numbers 1, …, n are placed on a circle clockwise in this order. Let σ denotes the shift which maps every element to the next position clockwise: σ(k) = k + 1 (mod n) and let σ^{(i)} be its ith iterate. (These are also referred to as rotations.) For the clarity of language we use the same symbol σ for the corresponding action on the vertex set and the edge set of Q_{n}. We will produce a σ-invariant edge partition.

We build, inductively on n, a family F_{n} of n-element sequences, starting for the case n = 1 with the family F_{1} = { ( }. To build F_{n+1} apply Operations A and B to every sequence F ∈ F_{n} and let F_{n+1} comprise the resulting sequences. Operation A: add ‘(’ to the right of F. Operation B: add ‘)’ to the right of F and throw away the resulting sequence if it does not contain free elements (i.e. if all its parentheses can be paired).

Proceeding in this way we obtain, for example,

F_{2} = { ( ( },
F_{3} = { ( ( (, ( ( ) }
F_{4} = { ( ( ( (, ( ( ( ), ( ( ) ( }.

It is easy to see that F_{n} is the set of all n-sequences beginning with ‘(’ which is a free element. (In particular, all right parentheses are paired.)

For any sequence F ∈ F_{n} we build the corresponding chain C_{F} in L(B_{n}) which has length t, where t is the number of free members of F. To obtain the (∗)-description of the ith element of C_{F}, i ∈ [t], we reverse in F the last i − 1 free parentheses and replace the ith free element (when counted from the right) by star ∗. Thus, for example, ‘( ( ) ( ( )’ gives ( ( ) ∗ ( ) and ∗ ( ) ) ( ) which correspond to the following chain in L(B_{6}):

({3, 6} ⊏ {3, 4, 6}) ⋖ ({3, 4, 6} ⊏ {1, 3, 4, 6}).

It is easy to see that every C_{F} is a symmetric chain. We claim that

D_{n} = {σ^{(j)}(C_{F}) : F ∈ F_{n}, j = 0, …, n − 1}

is the required edge partition.

We have to prove that for every element x = (A ⊏ B) in L(B_{n}) there are unique F ∈ F_{n} and j ∈ [0, n − 1] such that x ∈ σ^{(j)}(C_{F}). First we show how to find at least one such pair (F, j).

Step 1. Write x in the (∗)-representation. Step 2. Rotate the pattern to bring the star to position 1 and then identify all free parentheses. Clearly, if disregarding the paired elements, our sequence is ‘∗) …) ( … (.’ Step 3. Rotate again so that the first free left parenthesis identified in Step 2 (or the star itself if no ‘(’ is free) is moved to position 1. Let j be the number of positions that the star was moved anticlockwise by Steps 2 and 3 combined. Step 4. Replace the star and all free right parentheses identified in Step 2 by left parentheses. Let F be the resulting sequence.

Obviously, when we pair brackets in Step 4, we obtain the same sets of free/paired elements as in Step 2. This implies that F ∈ F_{n} as it starts with free ‘(’ and that x ∈ C_{F} as required. Here is an illustration for x = ({1, 6, 7} ⊏ {1, 4, 6, 7}) ∈ L(B_{8}):

Step 1: ) ( ( ∗ ( ) ) (
Step 2: ∗ ( ) ) ( ) ( (
Step 3: ( ( ∗ ( ) ) ( )  (and j = 1)
Step 4: ( ( ( ( ) ( ( )  (this is F)

The uniqueness of (F, j) can be established in different ways. One, which actually gives an alternative definition of D_{n}, is the following. Given the (∗)- representation of x, for 0 ≤ i ≤ n − 1 let g(i) = l_{i} − r_{i}, where l_{i} and r_{i} are respectively the number of left and right parentheses in the i positions preceding ‘∗’ clockwise. If x ∈ σ^{j}(C_{F}) then j is the smallest element in [0, n − 1] on which g achieves its maximum. Why? Just pair the brackets in the (∗)-representation of σ^{−j}(x) ∈ C_{F}, e.g.

( ( ( ) ( ) ) ( ) ( ∗ ) ( ) ) ( ) ,

and notice that any paired block (boxed regions) contributes 0 to g while any right-hand-sided part of it contributes a strictly negative value. Now, the maximum of g is the number of free left parentheses and this is achieved for first time when considering the segment preceding the star, as required.

But now, once that j has been identified, there trivially could not be two suitable F ’s.

For the remainder of this part let D_{n} denote the edge decomposition of B_{n} constructed above. It has the following properties.

Theorem 68 Let C = (A_{1} ⊏ … ⊏ A_{k}) be one of the chains in D_{n}. If A_{i+1} = A_{i} ∪ {a_{i}}, then the elements a_{1}, …, a_{k−1} are situated on the circle in this other anticlockwise and between a_{i} and a_{i+1} (anticlockwise) there is an even number of places. For each i ∈ [k − 3], there is an element (B ⊏ B′) belonging to a chain of D_{n} shorter than C such that

A_{i} ⊏ B ⊏ B′ ⊏ A_{i+3}.  (92)

Proof. Take the sequence F ∈ F_{n} giving rise to C. (We may assume j = 0.) The fact that in F every pair of consecutive free elements contains only paired brackets in between implies the firt part of the theorem.

To show the second claim, let F′ be the sequence F with the (i + 1)st free left bracket (if counted from the right) replaced by ‘)’ which is then paired with the (i + 2)nd free element:

F :  ( □ ( □ ( □ ( □ ( □ ( □
∗ . . . . . . . . . . A_{i}
∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A_{i+3}
F′ :  ( □ ( □ ( □ ) □ ( □ ( □
∗ . . . . . . . . . . B
∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B′

The new sequence corresponds to a chain of length k − 2 and its ith and i + 1st elements obviously satisfy the required property.

We define the complementary chain \overline{C} of a chain C by replacing every element by its complement, ie. if C = (A_{1} ⊂ … ⊂ A_{k}) then \overline{C} = (\overline{A_{k}} ⊂ … ⊂ \overline{A_{1}}).

Lemma 69 Two elements x_{1} = (A_{1} ⊏ B_{1}) and x_{2} = (A_{2} ⊏ B_{2}) of L(B_{n}) can belong simultaneously to D_{n} and \overline{D_{n}} only if n = 2k is even and \{|B_{1}|, |B_{2}|\} = {k, k + 1}.

Proof. Let i_{h} ∈ [n] be the element of B_{h} not in A_{h}, h = 1, 2, and let pairs (F, j) and (F′, j′) give rise to chains C, C′ ∈ D_{n} such that C′ contains x_{1} and x_{2} while C contains \overline{x_{1}} and \overline{x_{2}} respectively. Assume that j′ = 0 and x_{1} < x_{2}, ie. B_{1} ⊂ A_{2}.

In F′ i_{2} precedes i_{1} and we claim that F′ does not contain a free element between them. Indeed, if it be in the position y ∈ [n] then y ∈ \overline{B_{1}} and y ∉ \overline{A_{2}}, that is, σ^{−j}(y) must be a free element in F. But then σ^{−j}(y) must lie between σ^{−j}(i_{1}) < σ^{−j}(i_{2}). (In C the element \overline{x_{2}} comes before \overline{x_{1}}.) This contradiction (on one hand the elements i_{2}, y, i_{1} go clockwise, on the other—anticlockwise) proves the claim.

Thus all the elements between i_{2} and i_{1} are paired in F′; therefore B_{1} = A_{2} and there must be the same number of left and right parentheses in this interval. Considering \overline{x_{2}}, \overline{x_{1}} ∈ C we show the analogeous statement about the elements between i_{1} and i_{2} (if going clockwise), which clearly implies the claim.

16 APPLICATIONS OF THE PARTITION D_{N} 105

16 Applications of the Partition D_{n}

We would like to include here some applications of the edge partition D_{n} built in Theorem 67. Basically, we are inspired by known results where a symmetric vertex decomposition of B_{n} is used. We refer the reader to Section 3.4 of Anderson’s book [And87] for an exposition of a few results of this type. I am grateful to Ian Anderson for drawing my attention to some other applications not surveyed in his book.

16.1 On the Number of Antichains in L(B_{n})

Let us consider the following question: what is φ(L(B_{n})), the number of antichains in L(B_{n})? The computation of φ(B_{n}) is an old and difficult problem; a complicated asymptotic formula was established by Korshunov [Kor81].

Here we provide some rough estimates of φ(L(B_{n})) by applying ideas of Hansel [Han66] who showed that 2^{N} ≤ φ(B_{n}) ≤ 3^{N}, where N = w(B_{n}) = \binom{n}{⌊n/2⌋}.

Considering all possible subsets of the largest antichain of L(B_{n}) we obtain trivially φ(L(B_{n})) ≥ 2^{m}, where m = w(L(B_{n})) = ⌈n/2⌉\binom{n}{⌊n/2⌋}.

On the other hand, observe that an antichain A ⊂ L(B_{n}) is uniquely determined by the ideal Δ(A) = {x ∈ L(B_{n}) : ∃ a ∈ A x ≤ a}. Consider any C = (x_{1} ⋖ … ⋖ x_{l}) ∈ D_{n}. By Theorem 68 for 3 ≤ i ≤ l − 2 we can find y_{i} in a shorter chain with x_{i−2} < y_{i} < x_{i+2}. Knowing Δ(A) ∩ C′ for every C′ ∈ D_{n} shorter than C we know Δ(A) ∩ {y_{3}, …, y_{l−2}}. But then it is easy to check that only for at most 4 elements of C we are unable to deduce whether it is in Δ(A), and therefore Δ(A) ∩ C can assume at most 5 possible values. Considering consecutively the chains of D_{n} in some size-increasing order we conclude that φ(L(B_{n})) ≤ 5^{m}.

16.2 Orthogonal Partitions of L(B_{n})

Two chains in a poset P are called orthogonal if they have at most one common element. Two vertex chain partitions D and D′ are orthogonal if any C ∈ D is orthogonal to any C′ ∈ D′. A result of Shearer and Kleitman [KS79] (see [And87, Section 3.4]) asserts that there exist two orthogonal chain decompositions of B_{n} into \binom{n}{⌊n/2⌋} chains each.

What can be said about L(B_{n})? If n is odd, then D_{n} and \overline{D_{n}} are orthogonal by Lemma 69, where D_{n} is the decomposition built in Theorem 67.

Theorem 70 For odd n there is a pair of orthogonal symmetric chain decompositions of L(B_{n}).

Remark. Unfortunately, we do not know if the corresponding claim is true for even n.

The result of Baumert, McEliece, Rodermich and Rumsey [BMRR80] (for a proof, see [And87, Section 3.4.3] or [Bol86, Section 6]) states that posets admitting a pair of orthogonal decompositions satisfy the probabilistic form of Sperner’s theorem, which in our case, by Theorem 70, is the following.

Corollary 71 If two elements x and y in L(B_{n}), odd n, are chosen independently with arbitrary probability distribution (same for both elements) then P {x ≤ y} ≥ w(L(B_{n}))^{−1}.

16.3 A Storage and Retrieval Problem

Suppose we maintain a database with n records which we number from 1 to n and we wish to organize an efficient searching. We assume that we have queries Q_{1}, …, Q_{M} each of which we identify with the set of records satisfying it, that is, Q_{i} ⊂ [n] and these subsets are not necessarily distinct. One idea, see Ghosh [Gho75], is to find a sequence X of elements of [n] such that every Q_{i} occurs in X as a subsequence of consecutive terms so that every Q_{i} can be defined by a starting position in X and the size of Q_{i}.

In connection with this Lipski [Lip78] considered the following problem. Find the shortest sequence of elements of X = [n] such that X contains every A ⊂ [n] as a subsequence of |A| consecutive terms. He showed that s_{n}, the length of an optimal sequence, satisfies

(\frac{2}{πn})^{1/2} 2^{n} ≤ (1 + o(1))s_{n} ≤ (\frac{2}{π} ) 2^{n}.  (93)

As far as I know, these might be the best known bounds to date.

Here we consider a similar problem. Namely, we ask what is the value of t_{n}, the shortest length of a sequence X such that for every A ⊏ B ⊂ [n] the sequence X contains A as a subsequence of |A| consecutive terms preceded by x, where {x} = B \ A. Such a situation can happen if every query is a set with a selected point. For example, we search in a dictionary, the allowed queries are of the form “Find word” and the answer should give the entry where word is defined plus all relevant entries. Applying the ideas of Lipski [Lip78] we find the following upper and lower bounds.

Theorem 72

(\frac{n}{2π} )^{1/2} 2^{n} ≤ (1 + o(1))t_{n} ≤ (\frac{n}{π}) 2^{n}.  (94)

Proof. As the number of different pairs A ⊏ A ∪ {x} with |A| = ⌊n/2⌋ which can lie within a sequence of length m does not exceed m − ⌊n/2⌋ we conclude that

t_{n} ≥ ⌈n/2⌉\binom{n}{⌊n/2⌋} + ⌊n/2⌋

which implies the lower bound in (94) by Stirling’s formula.

On the other hand, associate with every chain C = (A_{1} ⊏ … ⊏ A_{q}) in D_{n} a sequence of elements of [n] which contains first the elements of A_{1} in any order which then are followed by a_{2}, …, a_{q}, where {a_{i}} = A_{i} \ A_{i−1}, i = 2, …, q.

Let [n] = S ∪ T be a partition of [n] into 2 parts of (nearly) equal sizes. Let ϕ_{1}, … ϕ_{k} be the sequences corresponding to a symmetric vertex decomposition of 2^{T}. Also, let ψ_{1}, …, ψ_{l} be the sequences corresponding to a symmetric edge decomposition of 2^{S}, each sequence being reversed.

Clearly, for every A ⊂ S there exists ϕ_{i} containing A as the first consecutive |A| terms and for every A ⊏ A ∪ {x} ⊂ T there exists ψ_{j} containing, at the end, A preceded by x.

Now consider the sequence

X = ψ_{1}ϕ_{1}ψ_{1}ϕ_{2} … ψ_{1}ϕ_{k}ψ_{2}ϕ_{1}ψ_{2}ϕ_{2} … ψ_{l}ϕ_{k}.

Take any A ⊂ [n] and x ∈ T \A. There is ψ_{i} containing x at the end followed by A∩T and ϕ_{j} containing A∩S as an initial subsequence. Therefore, X_{1} contains x followed by A. Interchanging S and T, we write a sequence X_{2} containing every pair A ⊏ A ∪ {x} with x ∈ S. The sequence X = X_{1}X_{2} is the required (and explicitly constructed) sequence. It is easy to see that the average size of a sequence corresponding to a chain of a symmetric vertex or edge decomposition of B_{n} is (\frac{1}{2} + o(1))n. Therefore,

t_{n} ≤ |X| ≤ 2 (\frac{1}{2} + o(1)) nkl

which gives the desired upper bound by Stirling’s formula.

16.4 One Numerical Problem

There exists a so called Audit Expert Mechanism which can be used to protect small statistical databases, see Chin and Ozsoyoglu [CO82]. To find an optimal mechanism the following problem has to be solved. Suppose we operate with n-tuples of non-zero real numbers a_{1}, …, a_{n} and we want to find what is the maximum possible number of subsets I ⊂ [n] such that a_{I} is equal to either 0 or 1. (Here and later we denote a_{I} = ∑_{i∈I} a_{i}.) The best possible bound of \binom{n+1}{⌊(n+1)/2⌋} was found by Miller, Roberts and Simpson [MRS91] and all extremal sequences were characterized by Miller and Sarvate [MS95]. Both papers make use of the existence of a symmetric chain decomposition of B_{n}.

Here, applying a symmetric chain decomposition of L(B_{n}), we can find K, the maximal possible number of elements (I ⊏ J) ∈ L(B_{n}) such that {a_{I}, a_{J}} = {0, 1}, over all real sequences a_{1}, …, a_{n}. Actually, we can allow zero entries for, as we will see later, this does not affect K. Apparently, this problem does not have such an application like that of the original problem, but it might be of some interest especially as an unexpected application of a symmetric chain decomposition of L(B_{n}).

The expression (a)^{i} is a shorthand for a repeated i times. Also we assume that all n-tuples have their entries ordered non-decreasingly.

Theorem 73 For n ≥ 2 we have

K = ⌈n/2⌉\binom{n}{⌊n/2⌋},  (95)

and this value is achieved for and only for the following sequences. For n = 2k, ((−1)^{k}, (+1)^{k}), ((−1)^{k−1}, (+1)^{k+1}) and ((−1)^{k−1}, 0, (+1)^{k}). For n = 2k + 1, ((−1)^{k}, (+1)^{k+1}).

Proof. Let m be the largest index for which a_{m} < 0. Define f : 2^{[n]} → 2^{[n]} by the formula

f(I) = I△[m] = (I \ [m]) ∪ ([m] \ I), I ⊂ [n].

One can easily check that I ⊂ J ⊂ [n] implies a_{f(I)} ≤ a_{f(J)}.

D_{n} can be viewed as a collection of symmetric chains in 2^{[n]}. Let X_{r} ⊏ … ⊏ X_{n−r} be one such chain. The sequence

a_{f(X_{r})}, …, a_{f(X_{n−r})}

is non-decreasing and therefore 0 and 1 can occur side by side there at most once. As every A ⊏ B is present in exactly one chain and f is a bijection preserving or reversing the ⊏-relation, K does not exceed the total number of chains, which gives the required upper bound.

A moment’s thought reveals that a necessary and sufficient condition for an n-tuple to be optimal is the following. If n = 2k + 1 then for every A ⊏ B ⊂ X, |A| = k, we have a_{f(A)} = 0 and a_{f(B)} = 1. If n = 2k then for every A ⊏ B ⊏ C ⊂ X, |A| = k − 1, among the numbers

a_{f(A)} ≤ a_{f(B)} ≤ a_{f(C)}  (96)

there is a 0 adjacent a 1.

This condition is fulfilled for the sequences mentioned in the statement. Indeed, let us consider ((−1)^{k}, (+1)^{k+1}), for example. Here m = k and for any A ⊏ B with |A| = k we have

a_{f(A)} = a_{A△[k]} = (−1)(k − s) + (k − s) = 0,  (97)

where s = |A ∩ [k]|. Similarly, a_{f(B)} = 1 so the sequence is optimal.

We claim that these are essentially the cases of the equality. Let us do the harder case n = 2k. If, for some i ≠ j, we have a_{i} ≠ ±1 and a_{j} ≠ ±1, then A ⊏ A ∪ {i} ⊏ A ∪ {i, j} with any A ∈ X^{(k−1)}, A ∌ i, j, obviously violates the condition. If, for exactly one i, we have a_{i} ≠ ±1, then considering A ⊏ A ∪ {i} ⊏ C we conclude that a_{f(A∪{i})} = 0 for any A ∈ (X \ {i})^{(k−1)}. Suppose a_{i} ≥ 0, for example. Then a_{f(A∪{i})} = k − j − 1 + a_{i} = 0, where j is the total number of elements equal to −1 (so 2k − 1 − j elements equal +1). If a_{i} = 0, then we have the third example mentioned in the theorem. If a_{i} ≥ 2 then j ≥ k + 1 and any sequence (96) with C ∌ i violates the condition. Finally, if |a_{i}| = 1 for every i then arguing as in (97) we deduce that we can have either k or k + 1 positive entries.

17 Characterization of Line Posets

For graphs we know that we can characterize line graphs in terms of forbidden induced subgraphs (Beineke [Bei68]) and we can reconstruct a connected graph G given L(G) except for L(G) = K_{3} when G is either K_{3} or K_{1,3}.

Here we ask ourselves when a given poset L is the line poset of some P and what information about P can be reconstructed from L(P). (Of course, it is implicitly understood that we operate with isomorphism classes of posets.) While for line graphs there are nine forbibben configuration, for line posets we have only two (or infinitely many, depending on how we look at it).

Note that L(P) cannot contain elements w, x, y, z such that w ⋖ y, x ⋖ y, w ⋖ z but x ⋖̸z; call this configuration N. Indeed, if y and z cover w they must be of the form (a ⋖ b), (a ⋖ c), where w = (d ⋖ a), some a, b, c, d ∈ P. Then the relation x ⋖ y implies that x = (e ⋖ a) which implies that x ⋖ z.

Also, L(P) cannot contain the configuration C_{n}, n ≥ 3, made of elements y and x_{1}, …, x_{n} such that x_{1} ⋖ y ⋖ x_{n} and x_{i} ⋖ x_{i+1}, for i ∈ [n − 1]. Indeed, suppose the contrary. Clearly, P contains elements z_{0} ⋖ z_{1} ⋖ … ⋖ z_{n} such that x_{i} = (z_{i−1} ⋖ z_{i}). But y covers the same element as x_{2} and is covered by the same element as x_{n−1}, so y = (z_{1} ⋖ z_{n−1}) and n = 3; but then y = x_{2}, which is a contradiction.

For a poset P let T (P) = (C, k, l, u) be the quadruple with C being a subposet of P spanned by the non-extremal elements, that is by {a ∈ P : ∃ b, c ∈ P, b < a < c} and k is the number of pairs (a ⋖ b) with a, b ∈ P \ C while the functions l, u : C → N_{0} are given by

l(a) = |{x ∈ P \ C : x ⋖ a}|,
u(a) = |{x ∈ P \ C : x ⋗ a}|,  a ∈ P.

It is easy to see that T (P) determines L(P) uniquely.

The following theorem states that the above examples provide a complete answer to our two questions.

Theorem 74 A poset L is isomorphic to L(P) for some P if and only if L contains neither configuration N nor any of C_{n}, n ≥ 3. Furthermore, T (P) determines L(P) and can be reconstructed from it.

Proof. Given a poset L without N or C_{n} let X be two disjoint copies of its vertex set, namely X = {x^{∧}, x^{∨} : x ∈ L}. Let x^{∧} ∼ y^{∨} if x ⋖ y; let x^{∧} ∼ y^{∧} if, for some s ∈ L, we have s ⋗ x and s ⋗ y; let x^{∨} ∼ y^{∨} if, for some s ∈ L, s ⋖ x and s ⋖ y.

We claim that ∼ is an equivalence relation. Indeed, if x^{∧} ∼ y^{∧} and y^{∧} ∼ z^{∧} then there are s, t ∈ L such that x, y ⋖ s and y, z ⋖ t. But then t must cover x for otherwise x, y, s, t would span a forbidden configuration. So x, z ⋖ t and x^{∧} ∼ z^{∧}. The remaining cases are equally easy.

Let \overline{x} denote the equivalence class of x ∈ X. Define the poset P (also denoted by L^{−1}(L)) on V = X/∼ = \{\overline{x} : x ∈ X\} by A < B, A, B ∈ V iff in L there exist y ≤ z with y^{∨} ∈ A and z^{∧} ∈ B. One can check that this is indeed an ordering. For example, to check its transitivity, let A < B and B < C, choose w ≤ x and y ≤ z in L with w^{∨} ∈ A, x^{∧}, y^{∨} ∈ B and z^{∧} ∈ C; then x^{∧} ∼ y^{∨} implies that w ≤ x ⋖ y ≤ z and A < C.

Let us show that \overline{x^{∧}} covers \overline{x^{∨}}. Assuming the contrary we find z ≥ y and w ≥ v in L with z^{∧} ∼ x^{∧}, y^{∨} ∼ w^{∧} and v^{∨} ∼ x^{∨}. By the definition of ∼, some t ∈ L covers both x and z, some s ∈ L is covered by both x and v and v ≤ w ⋖ y ≤ z—which implies that L contains some C_{n}, which is forbidden.

We claim that L ≅ L(P) via the map F which sends x ∈ L to (\overline{x^{∨}} ⋖ \overline{x^{∧}}). First note that F is an order preserving map: if x ⋗ y in L then \overline{x^{∨}} ∼ \overline{y^{∧}} which implies F (x) ⋗ F (y) as desired. Next, F is injective for if F (x) = F (y) then x^{∧} ∼ y^{∧} and x^{∨} ∼ y^{∨} which implies that for some w and z we have w⋖x⋖z and w ⋖ y ⋖ z; but as L does not contain configuration C_{3} we conclude that x = y. To show that F is surjective take any (A ⋖ B) ∈ L(P). As A < B, for some L-elements x ≤ y we have A = \overline{x^{∨}}, B = \overline{y^{∧}}. But it is easy to see that \overline{x^{∧}} ≤ \overline{y^{∧}}, which implies (A ⋖ B) equals (\overline{x^{∨}} ⋖ \overline{x^{∧}}) = F (x). Finally, if F (x) ⋖ F (y) then x^{∧} ∼ y^{∨} and x ⋖ y. This proves completely that L ≅ L(P).

In the second part it is enough to show that for any poset R we have T (R) ≅ T (P), where P = L^{−1}(L), L = L(R). To build a natural isomorphism H : C(R) → C(P) take, for any element a ∈ C(R), some b ⋖ a which exists as a is a non-extremal element of R. Now let H(a) = \overline{x^{∧}}, where x = (b ⋖ a) ∈ L and ∼ is as above. To show that H is well defined, let b′ be another choice of b and denote y = (b′ ⋖ a). Let c be an element covering a. Then (a ⋖ c) covers in L both x and y, so by the definition of P we have x^{∧} ∼ y^{∧}. Also, H(a) ∈ P is not extremal as

\overline{(b ⋖ a)^{∨}} < H(a) < \overline{(a ⋖ c)^{∧}}.

Next, H is an order-preserving bijection. Indeed, let a ⋗ b in C(R). Choose c ⋖ b. Then H(a) = \overline{(b ⋖ a)^{∧}} and H(b) = \overline{(c ⋖ b)^{∧}}. But (c ⋖ b)^{∧} ∼ (b ⋖ a)^{∨} and we have H(a) > H(b) by the definition of the order on P. To show that H is injective choose any a, a′ ∈ C(R). Then H(a) = H(a′) implies that y = (c⋖a)^{∧} ∼ y′ = (c′ ⋖a′)^{∧}, some c, c′ ∈ R. Therefore there is x ∈ L covering both y and y′ which implies a = a′ in R as required. To establish the surjectivity of H consider x = \overline{(a ⋖ b)^{∨}} ∈ C(P), for example. Observe first that a ∈ R is not extremal. Indeed, take any y ∈ P covered by x; as we have already shown any pair y ⋖ x is of the form \overline{(c ⋖ d)^{∨}} ⋖ \overline{(c ⋖ d)^{∧}} which implies d = a and c ⋖ a. Now H(a) = \overline{(c ⋖ a)^{∧}} = x as required. Again, any two adjacent elements of C(P) can be represented as \overline{(a ⋖ b)^{∨}} ⋖ \overline{(a ⋖ b)^{∧}} and then they are the images of two adjacent elements, a ⋖ b of C(R), which implies that C(P) ≅ C(R).

Finally, as P and R give rise to naturally isomorphic line posets, in the sense that

F (a ⋖ b) = (\overline{(a ⋖ b)^{∨}} ⋖ \overline{(a ⋖ b)^{∧}}) = (H(a) ⋖ H(b)) , a, b ∈ C(R),

our mapping H preserves k, l and u, which are naturally reconstructible from the line poset.

Part IV

Enumeration Results for Trees

18 Introduction

The notion of a tree and its different extensions to hypergraphs play an important role in discrete mathematics and computer science. We will dwell upon the following, rather general, definition suggested independently by Dewdney [Dew74] and Beineke and Pippert [BP77].

Let us agree that the vertex set is [n] = {1, …, n}. Fix the edge size k and the overlap size m, 0 ≤ m ≤ k − 1. We refer to k-subsets and m-subsets of [n] as edges and laps respectively. A non-empty k-graph without isolated vertices is called a (k, m)-tree if we can order its edges, say E_{1}, …, E_{e}, so that for every i, 2 ≤ i ≤ e, there is i′, 1 ≤ i′ < i, such that |E_{i} ∩ E_{i′}| = m and (E_{i} \ E_{i′}) ∩ (∪_{j=1}^{i−1}E_{j}) = ∅. In other words, we start with a single edge and can consecutively affix a new edge along an m-subset of an existing edge.

Thus, a (k, m)-tree with e edges has n = e(k − m) + m vertices and its edges cover f = e (\binom{k}{m} − 1) + 1 laps. For example, a (k, 0)-tree consists of disjoint edges.

The problem of counting (m + 1, m)-trees which are known in the literature as m-trees, received great attention and was completely settled by Beineke and Pippert [BP69] and Moon [Moo69]. This extends the celebrated theorem of Cayley [Cay89] as, clearly, 1-trees correspond to usual (Cayley) trees. Later, different bijective proofs for m-trees appeared as well, see [RR70, Foa71, GI75, ES88, Che93].

Here we enumerate (k, m)-trees. In fact, a considerable difficulty was to guess the right formula. When we had a plausible conjecture, we tried to prove it inductively by writing a recurrence relation. We were rather fortunate: the result reduced to the identity proved by Beineke and Pippert [BP69, Lemma 2]. This enabled us to write a short inductive proof, published in [Pik99c], which is presented in Section 19.

Of course, a bijective proof (that is, a correspondence between the set of trees to count and some simple set) is a far more satisfactory answer. (For example, a bijective proof may allow us to generate one by one all trees or to count the number of trees satisfying some given property.) In Subsection 20.2 we exhibit an explicit bijection between the set of rooted vertex labelled trees of given size and a trivially simple set; it is based on the ideas of Foata [Foa71] which are presented in Subsection 20.1. The knowledge of the actual formula was essential, as otherwise we would have had little idea what and how to biject.

In fact, this method (based on Foata’s bijection) can be applied to enumerate bijectively other tree-like structures. For example, we can enumerate so called k-gon trees, a structure studied in [CL85, Whi88, Pen93, KT96]. In order not to repeat the same portions of proof twice, we present a more general result including both (k, m)-trees and k-gon trees as partial cases.

In Subsection 20.3 we consider the question whether our bijection can count edge labelled trees. We present a construction for 2-graphs only, which in fact answers a question posed by Cameron [Cam95]. This question was motivated by the possibility that such a bijection might simplify some of his enumeration results (or proofs) from [Cam95]. However, although we answered Cameron’s question, we were not able to improve [Cam95]. Please refer to Subsection 20.3 for further details.

19 Inductive Proof

Let T_{km}(e) denote the number of distinct (k, m)-trees on [n] with e edges, n = e(k −m)+m, and let R_{km}(e) count the trees rooted at the lap [m], that is, those trees for which [m] is covered by some edge.

Theorem 75 Given integers k, m, e with 0 ≤ m ≤ k − 1 and e ≥ 1, let n = e(k − m) + m, l = \binom{k}{m} and f = e(l − 1) + 1. Then the number of different (k, m)-trees on [n] equals

T_{km}(e) = \frac{n!f^{e−2}}{e!m! ((k − m)!)^{e}} .  (98)

Proof. As in Beineke and Pippert [BP69], to prove the theorem, we write down a recurrence relation for T_{km}(e) and then verify that (98) does satisfy the relation. Let us agree that T_{km}(0) = R_{km}(0) = 1.

Counting in two different ways the number of pairs (H, L), where H is a (k, m)-tree on [n] rooted at L ∈ [n]^{(m)}, we obtain

\binom{n}{m}R_{km}(e) = f · T_{km}(e).  (99)

Next, consider the following method for constructing trees. Select an edge E ∈ [n]^{(k)} and label by L_{1}, …, L_{l} the laps of E. Represent e − 1 as a sum of l non-negative integers, e − 1 = e_{1} + … + e_{l}. Partition [n] \ E into sets X_{1}, …, X_{l} of sizes e_{1}(k −m), …, e_{l}(k −m) respectively. On each L_{i} ∪X_{i} build a (k, m)-tree H_{i} rooted at L_{i}, i ∈ [l]. Clearly, the union of all H_{i}’s plus the edge E forms a (k, m)-tree and every such tree is obtained exactly e times. Therefore, by (99), we obtain

eT_{km}(e) = \binom{n}{k} ∑_{e} \frac{(n − k)!}{(e_{1}(k − m))! … (e_{l}(k − m))!} ∏_{i=1}^{l} R_{km}(e_{i})
= \frac{n!}{k!} ∑_{e} ∏_{i=1}^{l} \frac{m!(e_{i}(l − 1) + 1)T_{km}(e_{i})}{(e_{i}(k − m) + m)!},  (100)

where ∑_{e} denotes the summation over all representations e − 1 = e_{1} + … + e_{l} with non-negative integer summands.

Clearly, formula (98) gives correct values for e = 0. Also, the substitution of (98) into the both sides of (100) gives (after routine cancellations)

l(e(l − 1) + 1)^{e−2} = ∑_{e} \frac{(e − 1)!}{e_{1}! … e_{l}!} ∏_{i=1}^{l}(e_{i}(l − 1) + 1)^{e_{i}−1}.

The last identity (in slightly different notation) was established by Beineke and Pippert [BP69, Lemma 2], which proves our theorem by induction.

Corollary 76 The number of vertex labelled m-trees on n vertices, n > m ≥ 1, is T_{m+1,m}(n − m) = \binom{n}{m}(mn − m^{2} + 1)^{n−m−2}.

20 Bijective Proofs

20.1 Foata’s Bijection

Given disjoint finite sets A, B, C and a surjection γ : B → A, a function f : A → B ∪ C is called cycle-free if for every b ∈ B the sequence (f ∘ γ)^{i}(b) eventually terminates at some c ∈ C. Foata [Foa71, Theorem 1] exhibited a bijection between F (A, B, C, γ), the set of cycle-free functions, and the set of functions g : A → B ∪ C such that g(a_{1}) ∈ C, some beforehand fixed a_{1} ∈ A; this implies

|F (A, B, C, γ)| = |C|(|B| + |C|)^{|A|−1}.  (101)

We briefly describe a simpler construction than that in [Foa71]. Fix some ordering of A. Let f ∈ F (A, B, C, γ). Let Z = (z_{1}, …, z_{s}) denote the increasing sequence of the elements in A \ γ(f(A)). (For convenience we assume that γ(c) = c, c ∈ C.) We build, one by one, s sequences δ_{1}, …, δ_{s} composed of elements in B ∪ C. Having constructed sequences δ_{1}, …, δ_{i−1}, let m_{i} ≥ 0 be the smallest integer such that (f ∘ γ)^{m_{i}}(f(z_{i})) either belongs to C or occurs in at least one of δ_{1}, …, δ_{i−1}. We define (mind the order)

δ_{i} = ((f ∘ γ)^{m_{i}}(f(z_{i})), (f ∘ γ)^{m_{i}−1}(f(z_{i})), …, f(z_{i})) .  (102)

One can check that Z is non-empty if A is, every m_{i} exists, and δ, the juxtaposition product of the s sequences δ_{1}, …, δ_{s}, contains |A| elements. (In fact, δ is but a permutation of (f(a))_{a∈A}.) The obtained sequence δ of |A| elements of B ∪ C, which starts with an element of C, corresponds naturally to the required function g : A → B ∪ C.

Conversely, given g (or δ), we can reconstruct Z which consists of the elements of A\γ(g(A)). Then, exactly s = |Z| times, an element of δ either belongs to C or equals some preceding element. These s elements mark the beginnings of δ_{1}, …, δ_{s}. Now we can restore the required f by (102). To establish (101) completely, one has to check easy details.

20.2 H-Built-Trees

Adopting the ideas of Foata [Foa71], we present a bijective proof of (98). Our method can enumerate some other tree-like structures. For example, we can find a bijection for vertex labelled k-gon trees (also known as cacti or trees of polygons), a structure that appears in [CL85, Whi88, Pen93, KT96].

We define a k-gon tree inductively. A k-gon (that is, a k-cycle) is a k-gon tree. A k-gon tree with g + 1 k-gons is obtained from a k-gon tree with g k-gons by adding k − 2 new vertices and a new k-gon through these vertices and an already existing edge. Thus, a k-gon tree is a (usual) 2-graph; if we have g k-gons, then it has e = g(k − 1) + 1 edges and n = g(k − 2) + 2 vertices.

In order not to repeat the same portions of proof twice, we present the following, more general, result which includes (k, m)-trees and k-gon trees as partial cases.

Let H be any m-graph on [k]. An H-built-tree (T, {H_{1}, …, H_{e}}) consists of a usual (k, m)-tree T with edges E_{1}, …, E_{e} plus H-graphs H_{i} on E_{i}, i ∈ [e], such that if E_{i} ∩ E_{j} is a lap (that is, has size m), then it is an edge of both H_{i} and H_{j}, for any distinct i, j ∈ [e]. Let n = e(k − m) + m be the total number of vertices and let

f = |∪_{i∈[e]}E(H_{i})| = e(l − 1) + 1,

where l = e(H). Also, let R_{H} be the set of distinct H-graphs on [k] rooted at [m], that is, containing [m] as an edge. Clearly,

|R_{H}| = \frac{k!l}{\binom{k}{m}|Aut(H)|}.

An H-built-tree is rooted on an m-set L if L ∈ ∪_{i∈[e]}E(H_{i}).

Theorem 77 There is a bijection between the set Y of H-built-trees on [n] rooted at [m] and the set

Z = F (A, B, C, γ) × ∏_{i=1}^{e} (X_{i} × R_{H}) ,

where A = [e], B = [e] × [l − 1], C = {[m]}, γ is the coordinate projection B → A, and X_{i} = [\binom{(k−m)(e−i+1)−1}{k−m−1}]. In particular,

|Y | = f^{e−1} (\frac{k!l}{\binom{k}{m}|Aut(H)|})^{e} ∏_{i=1}^{e} \binom{(k − m)(e − i + 1) − 1}{k − m − 1}.

Proof. Given an H-built-tree T rooted at [m], order its edges E_{1}, …, E_{e} so that [m] ∈ E(H_{1}) and each E_{i}, i ∈ [2, e], shares a lap with some E_{j}, j < i.

Correspond an edge E_{i} to the lap g′(E_{i}) = E_{i} ∩ ∪_{j=1}^{i−1}E_{j}, 2 ≤ i ≤ e. (We agree that g′(E_{1}) = [m].) Call the set f(E_{i}) = E_{i} \ g′(E_{i}) the free part of E_{i}; the free parts partition [n] \ [m]. Clearly, these definitions of g′ and f do not depend on the particular ordering.

Relabel the edges by D_{1}, …, D_{e} so that d_{i} = min f(D_{i}) increases; let H′_{i} denote the corresponding H-graph on D_{i}. Label, in the colex order, all edges (laps) of H′_{i} but g′(D_{i}) ∈ E(H′_{i}) by (i, j), j = 1, …, l − 1. Note that now we have indexing of the edges of T by A, namely (D_{i})_{i∈A}, and of the laps of T by B ∪C. Let g : A → B ∪C be the map corresponding to g′. A moment’s thought reveals that g is cycle-free.

Repeat the following for i = 1, …, e. Index, in the colex order, the (k − m − 1)-subsets of (∪_{j=i}^{e}f(D_{j})) \ {d_{i}} by the elements of X_{i} and let x_{i} ∈ X_{i} be the index corresponding to f(D_{i}) \ {d_{i}}. Consider the bijection h : D_{i} → [k] such that h is monotone on g′(D_{i}) and f(D_{i}) which are respectively mapped onto

[m] and [m + 1, k]. Let r_{i} ∈ R_{H} be the image of H′_{i} under h.

Now, (g, x_{1}, r_{1}, …, x_{e}, r_{e}) ∈ Z is the ‘code’ of T ∈ Y.

Conversely, given an element (g, x_{1}, r_{1}, …, x_{e}, r_{e}) ∈ Z we can consecutively reconstruct the sequence (d_{i}, f(D_{i})), i = 1, …, e. Indeed, d_{i} is the smallest element of V = [n]\((∪_{j=1}^{i−1}f(D_{j}))∪[m]) while f(D_{i})\{d_{i}} is the x_{i}th (k−m−1)- subset of V \ {d_{i}}. For i ∈ A with g(i) ∈ C, we have D_{i} = [m] ∪ f(D_{i}) and (knowing g′(D_{i}) = [m] and f(D_{i})), we can determine H′_{i} from r_{i}; then we can recover the lap corresponding to (i, j) ∈ B as the jth lap of E(H′_{i}) \ {[m]}, j ∈ [l − 1].

Likewise, we can reconstruct all information about D_{i} for any i ∈ A with g(i) being already associated with a lap. As f is cycle-free, all edges are eventually identified, producing T ∈ Y.

A plain verification shows that we have indeed a bijective correspondence between Y and Z.

It is trivial to check that if a union of K_{k}^{m}-graphs can be formed into a K_{k}^{m}-built-tree, then the latter is uniquely defined. Hence, the number of vertex labelled (k, m)-trees equals the number of K_{k}^{m}-built-trees. Now, |R_{K_{K}^{m}}| = 1, |Y | = R_{km}(e), and we can easily deduce formula (98).

Similarly, k-gon trees are in bijective correspondence with C_{k}-built-trees. We have |R_{C_{k}}| = (k − 2)!, so we obtain that there are

(g(k − 1) + 1)^{g−1}((k − 2)!)^{g} ∏_{i=1}^{g} \binom{(k − 2)(g − i + 1) − 1}{k − 3}

rooted k-gon trees with g k-gons, which implies the following result.

Corollary 78 The number of vertex labelled k-gon trees with g k-gons is

\frac{(g(k − 2) + 2)!(g(k − 1) + 1)^{g−2}}{2(g!)} , k ≥ 3.

20.3 Edge Labelled Trees

Cameron [Cam95] enumerates certain classes of what is called there two-graphs: reduced, 5-free, and (5, 6)-free and presents their connections to Coxeter groups of graphs. Please refer to his work for all definitions and details. Also, he defines, for a given (Cayley) tree T, the equivalence relation ≅ on its edges which is the smallest one such that two edges are related if they intersect at a vertex of degree 2 in T. For example, T is series-reduced (that is, T does not contain a vertex of degree 2) if and only if ≅ is the identity relation.

Cameron had to count the number S_{n} of trees with n edges with labelled ≅-classes. He found the following formula ([Cam95, Proposition 3.5(a)]):

S_{n} = ∑_{k=1}^{n} S(n, k)\frac{1}{k + 1} ∑_{j=0}^{k−1}(−1)^{j}\binom{k + 1}{j}\binom{k − 1}{j}j!(k − j + 1)^{k−j−1}, (103)

where S(n, k) is the Stirling number of the second kind: the number of partition of an n-set into k non-empty parts. The sequence (S_{n}) starts as 1, 1, 2, 8, 52, … and probably cannot be represented in a simple form but, of course, one can try to simplify (103).

Cameron [Cam95] asks the following question.

Problem 79 (Cameron) Describe a constructive bijection between edge labelled trees and edge Prüfer codes, not going via vertex labellings. Describe the equivalence relation ≅ in terms of this code.

The motivation for the problem was apparently that such a code might simplify (103). Although we answer here this question, we have not so far been able to simplify (103) or its proof from [Cam95]. But anyway, let us describe our construction. Of course, we use Foata’s [Foa71] bijection for cycle-free functions.

Let e_{1}, …, e_{n} be the edges. Suppose e_{1} = {a, b}; this edge will play a special role. Let A = B = {e_{2}, …, e_{n}}, C = {a, b} and γ : B → A be the identity function. Let us correspond an f ∈ F (A, B, C, γ) to a given tree T. Each edge e can be connected to e_{1} by the unique path in T. If e is incident to e_{1}, then let f(e) be equal to their common vertex; otherwise, let f(e) be the first edge on the path from e to e_{1}. This gives a correspondence between twice the number of edge-labelled trees (we can label the two vertices of e_{1} by a and b in two different ways) and F (A, B, C, γ). Foata’s bijection shows that |F (A, B, C, γ)| = |C|(|A| + |C|)^{|A|−1}, which implies, as desired, that the number of edge-labelled trees with n edges is (n + 1)^{n−2}.

Of course, the code is rather simple; we describe briefly only one direction. A code δ is a sequence of length n − 1 consisting of elements in {a, b, e_{2}, …, e_{n}} and staring with a or b. The set Z ⊂ {e_{2}, …, e_{n}} of edges which do not occur in the sequence consists of leaves. (If a or b does not occur, then e_{1} is also a leaf.) Clearly, an element of δ equals either a or b or some previously occurring element exactly z = |Z| times. Cut δ before each such element; we have z pieces δ_{1}, …, δ_{z}. Append the ith element z_{i} of Z to the end of δ_{i} to obtain δ′_{i}, i ∈ [z].

The reversed sequence δ′_{i} describes the initial segment P′_{i} of the path P_{i} from the element z_{i} ∈ Z to e_{1} until it hits e_{1} or some previous path P_{j}, i ∈ [z]. Clearly, this determines some tree.

This bijection corresponds to every edge-labelled tree two codes, one starting with a and the other—with b. To make this correspondence one-to-one, we consider only a half of the codes, e.g. those starting with a.

How can we read the ≅-relation from δ? First, let ≅′ be the minimal equivalence relation on {a, b, e_{2}, …, e_{n}} such that e_{i} ≅′ e_{j} if e_{i} and e_{j} intersect at a vertex of degree 2, 2 ≤ i < j ≤ n, and x ≅′ e_{i} if x is a degree-2 vertex incident to e_{i}, x ∈ {a, b}, i ∈ [2, n]. (Informally, we cut e_{1} in its middle and take the usual ≅-relation on the both created components separately.) Clearly, ≅ is obtained from ≅′ by identifying a and b into a single element e_{1}, so let us indicate how to determine the latter relation.

Take any maximal contiguous subsequence S ⊂ δ consisting of elements that occur in δ exactly once. Clearly, S lies entirely within some δ_{i} and S ∪ {y} is a ≅′-equivalence class, where y is the symbol following S in δ′_{i}. Conversely, it is easy to check that all non-trivial ≅′-classes are obtained this way, as required.

This answers Problem 79. Unfortunately, I do not see how this description can simplify Cameron’s formula (103). Remark. We do not know any bijection enumerating edge labelled (k, m)-trees for k ≥ 3.

Part V

Large Degrees in Subgraphs

21 Introduction

All research carried in this part revolves around the following conjecture of Erdos [Erd81] which is disproved here.

Erdős [Erd81], see also e.g. [Chu97, Erd99], conjectured that for n ≥ 3 any graph with \binom{2n+1}{2} − \binom{n}{2} − 1 edges is a union of a bipartite graph and a graph with maximum degree less than n. This value arises from the consideration of P_{n+1,n} which does not admit the above representation. (P_{m,n} = K_{m} + E_{n} has m + n vertices of which m vertices are connected to every other vertex.)

In the arrowing notation the latter statement reads “P_{n+1,n} → (K_{1,n}, C_{odd})”: for any blue-red colouring of the edge-set of P_{n+1,n} we necessarily have either a blue star K_{1,n} or a red cycle of odd length. (By C_{odd} we denote the family of odd cycles.) Thus the conjecture states that r̂(K_{1,n}, C_{odd}) = e(P_{n+1,n}) and, if true, would give the same value for the size Ramsey number r̂(K_{1,n}, K_{3}), since certainly r̂(K_{1,n}, K_{3}) ≥ r̂(K_{1,n}, C_{odd}) and in fact P_{n+1,n} → (K_{1,n}, K_{3}).

We show, however, that both these size Ramsey numbers grow as n^{2} plus a term of order n^{3/2}. (Actually, the conjecture fails for all n ≥ 5.) More precisely, our main result is the following.

Theorem 80

r̂(K_{1,n}, K_{3}) < n^{2} + \sqrt{2} n^{3/2} + n,  for n ≥ 1,  (104)
r̂(K_{1,n}, C_{odd}) > n^{2} + 0.577 n^{3/2},  for sufficiently large n. (105)

In [FRS97, Section 1] it is asked whether the conjecture is true for graphs with (at most) m vertices. Faudree (for a proof see [ERSS96]) showed this is the case for m = 2n + 1. Our construction can beat P_{n+1,n} on 3n + 1 vertices. Perhaps P_{n+1,n} is extremal for graphs with 2n plus few more vertices, but even for 2n + 2 vertices we do not know whether this is true.

Some previous attempts to prove Erdős’ conjecture resulted in new interesting directions of research; here we investigate also some of these questions.

Erdős and Faudree [EF99] consider the related problem of determination of the minimal size of a graph G such that if G is a union of two graphs, one having maximal degree less than n, then the other contains all odd cycles C_{m} with 3 ≤ m ≤ n − 3. Here we demonstrate a graph G of size (1 + ε)n^{2}, for any given constant ε > 0, such that, for any blue-red colouring of G without a blue K_{1,n}, we have red cycles of all lengths (odd and even) between 3 and cn, where c = c(ε) > 0 does not depend on n.

For positive integers n, k, j with k ≥ j, Erdős, Reid, Schelp and Staton

[ERSS96] consider the property M(n, k, j) which is defined as follows. A graph G belongs to M(n, k, j) if it has n+k vertices and for every (n+j)-set A ⊂ V (G) we have Δ(G[A]) ≥ n. (That is, the maximal degree of the subgraph of G spanned by A is at least n.) The problem is to compute

m(n, k, j) = min{e(G) : G ∈ M(n, k, j)}.

Erdős et al [ERSS96, Conjecture 1] conjectured that for any n ≥ k ≥ j ≥ 1 and n ≥ 3, we have

m(n, k, j) = (k − j + 1)n + \binom{k − j + 1}{2}.  (106)

This value arises from the consideration of P_{k−j+1,n}⊔E_{j−1}. Erdős et al [ERSS96, Theorem 3] proved that (106) is true if j = 1 or if j ≥ 2 and

n ≥ max (j(k − j), \binom{k−j+2}{2}) .  (107)

In Section 23 we demonstrate a constructive counterexample to (106) if n ≤ (j − 2)(k − j). On the other hand, we show that the formula (106) is true if

n ≥ max ((j + \frac{1}{2}) (k − j) + \frac{j+k}{4j−2}, 14) ,

which is an improvement on (107) for j ⪅ k/3. This shows that the value j(k−j) is roughly the threshold on n when the obvious construction suggesting (106) fails to be extremal. Some other constructions are presented.

Another function whose study was motivated by Erdős’ conjecture is as follows. Let B(n, m) consist of all graphs such that, for any partition V (G) = A ∪ B, either Δ(G[A]) ≥ n or Δ(G[B]) ≥ m (or both). We are interested in the bisplit function b(n, m) = min{e(G) : G ∈ B(n, m)}. Clearly, b(n, n) = r̂(K_{1,n}, C_{odd}) is precisely the function investigated in Erdős’ conjecture, which was the original motivation for introducing the ‘off-diagonal’ numbers b(n, m).

In Subsection 24.1 we present a simple argument giving a lower bound on b(n, m), any n, m, and a construction of G ∈ B(n, m) (which obviously gives an upper bound) which together compute the function asymptotically when m = min(n, m) is large:

b(n, m) = 2nm − m^{2} + o(m)n.  (108)

Concerning small values of m, not much is known. Of course, the bounds of Subsection 24.1 are applicable here, but the error term is not negligible if m is bounded. Namely, we obtain that, for any fixed m ≥ 1, the numbers b(n, m), n ∈ ℕ, lie between two functions linear in n with slopes 2m + 1 and 2m + \sqrt{2m} + \frac{5}{2}.

We prove that b(n, 1) = 4n − 2 for n ≥ 8 (and characterize all extremal graphs) and that b(n, 2) = 6n + O(1). As the reader will see the proofs are rather lengthy and require consideration of many cases. This indicates that the computation of lim_{n→∞} b(n, m)/n for any fixed m (if the limit exists) is perhaps a hard task.

22 Triangle-vs-Star Size Ramsey Number

Here we will prove the bounds on r̂(K_{1,n}, K_{3}) stated in the introduction.

22.1 Upper Bound

Proof of (104). We provide an explicit construction of a (K_{1,n}, K_{3})-arrowing graph G.

Take any representation n = k_{1} + … + k_{m} and let G be the disjoint union of P_{k_{i},n}, i ∈ [m], plus a vertex x connected to everything else. Consider any bluered colouring of E(G) without a blue K_{1,n}. Among n(m+1) edges incident to x there are at least mn+1 red ones. By the pigeon-hole principle, x sends at least n + 1 red edges to some P_{k_{j},n}, say {x, y_{i}}, i ∈ [0, n], of which at least one must be incident to a vertex of K_{k_{j}} ⊂ P_{k_{j},n}, say y_{0}. But of n edges {y_{0}, y_{i}} ∈ E(G), i ∈ [n], one is necessarily red and creates a red triangle whose third vertex is x. Hence, G → (K_{1,n}, K_{3}).

We have e(G) = (m + n + 1)n + ∑_{i∈[m]} \binom{k_{i}}{2}. To minimize e(G) we take the k_{i}’s nearly equal; so they are essentially uniquely determined by m. Any value of m we choose will give some upper bound for r̂(K_{1,n}, K_{3}). Choose m so that n = 2m^{2} + r, where |r| ≤ 2m. So, for example, when n = 2m^{2} − 2m we could choose either m or m − 1. We believe, though we do not prove, that such a choice of m is optimal. The verification of (104) is now best split into four cases. For example, for 0 ≤ r ≤ m we have m − r times k_{i} = 2m and r times k_{i} = 2m + 1. Routine simplifications show that

2n^{3} − (e(G) − n^{2} − n/2 − r/2)^{2} = 3m^{2}r^{2} + 2r^{3} ≥ 0,

which implies (104). The other cases can be verified similarly.

One can check that the bound (104) gives strictly better values than \binom{2n+1}{2}− \binom{n}{2} for all n ≥ 6. In fact, Erdős’ conjecture fails also for n = 5 when the representation n = 2 + 3 produces a graph with 44 edges.

We do not know any example beating our construction, which therefore might be an extremal one, but we do not dare to make any conjecture yet. It is surprising that a counterexample was not found earlier. An explanation might be that P_{n+1,n} is perhaps extremal among all (K_{1,n}, C_{odd})-arrowing graphs with few vertices; as shown by Faudree (for a proof see [ERSS96]) this is the case for graphs of order 2n + 1. Note that we can beat P_{n+1,n} using 3n + 1 vertices for n ≥ 5: take m = 2 in our construction.

22.2 Lower Bound

In this section we suppose on the contrary to (105) that there is a (K_{1,n}, C_{odd})- arrowing graph G with at most n^{2} + 0.577 n^{3/2} edges and try to derive a contradiction for large n.

Instead of 2-colourings of E(G) we find it more convenient to operate with 2-partitions of V (G). Thus our assumption on G states that

max {Δ(G[A]), Δ(G[B])} ≥ n

for any partition V (G) = A ∪ B.

The following simple argument, which we call the greedy algorithm, shows that any A ⊂ V (G) spans at least n(n − |\overline{A}| + 1) edges, where \overline{A} = V (G) \ A.

Indeed, inductively let x_{i} be any vertex (if exists) of degree at least n in G[A\{x_{1}, …, x_{i−1}}]. Let X = {x_{1}, …, x_{k}} ⊂ A be the set eventually obtained. By definition, Δ(G[A \ X]) < n. But then \overline{A} ∪ X contains at least n + 1 vertices (to allow a vertex of degree n), that is, k ≥ n − |\overline{A}| + 1, and the claim follows.

Taking A = V (G) we obtain e(G) ≥ n^{2} +n. We will add an n^{3/2}-term to this trivial bound by using a probabilistic argument. But before we can apply it, we have to fiddle a lot with the greedy algorithm in order to gain some structural information about G.

Let us introduce some notation first. By d_{A}(x) = |A ∩ Γ(x)| we denote the number of neighbours of x lying in A, x ∈ V (G), A ⊂ V (G). Also let L = {x ∈ V (G) : d(x) ≥ n}, l = |L| − n and e(G) = n^{2} + c_{g}n^{3/2}. Thus we assume that c_{g} ≤ 0.577 and in fact, by adding edges to G, that c_{g} = 0.577+o(1).

Lemma 81 l ≤ c_{g}n^{1/2} + O(1).

Proof. Apply a modified greedy algorithm. Set initially A = C = ∅ and B = V (G). These three sets will always partition V (G).

Repeat the following as long as possible or until |A| = n + 1. Take a vertex x ∈ B (if exists) with d_{B}(x) ≥ n and move it to A; colour aqua all edges connecting x to B. Then for every such x do the n-check, that is, move to C all vertices in B ∩ L whose B ∪ C-degree is now smaller than n, that is, equals n−1. (Thus before we proceed with another x we ensure that a vertex z ∈ L\A belongs to B if and only if d_{B∪C}(z) ≥ n.)

When we stop we have a+c ≥ n+1, where a, b, c are the cardinalities of the eventual sets A, B, C. Indeed, if a < n+1 then Δ(G[B]) < n so Δ(G[A∪C]) ≥ n and the claim follows.

The number of aqua edges is e_{a} ≥ an. Call non-aqua edges incident to C cyan. Every vertex in C is incident to exactly n − 1 cyan edges; hence we have e_{c} ≥ c(n − 1) − \binom{c}{2} cyan edges.

By applying our usual greedy algorithm to B∪C we obtain that there is a set Y = {y_{1}, …, y_{n+1−a}} ⊂ B ∪C such that each y_{i} has at least n neighbours in the complement of A ∪ {y_{1}, …, y_{i−1}}. Clearly, Y must be disjoint from C, that is, Y ⊂ B. We have e_{y} ≥ (n + 1 − a)n edges between Y and C ∪ B; colour all these edges yellow. (Some edges may be yellow and cyan simultaneously.) Finally, each vertex in R = L∩(B \Y) has degree in B ∪C at least n (otherwise it would have been moved to C earlier). Hence R is incident to e_{r} ≥ r(n − |Y | − c) − \binom{r}{2} edges lying within B \ Y, where r = |R|; call them red edges.

We claim that c = o(n). Suppose not. As e_{a} + e_{y} > n^{2}, the number of cyanonly edges is o(n^{2}) and the average yellow-degree of x ∈ C is n + o(n); hence |Y | ≥ n+o(n). Now |C| ≥ |Y | because a+c ≥ n+1 = a+|Y |, so |C| ≥ n+o(n). But C ∪ Y ⊂ L and |L| ≤ 2n + o(n) by the handshaking lemma. Therefore c = n + o(n), a = o(n), r = o(n) and all but o(n^{2}) edges lie between C and Y. But consider partition V (G) = V_{1} ∪ V_{2} obtained by placing in V_{1} all of A ∪ R, n/3 vertices from C, n/3 vertices from Y and all (= o(n)) vertices from C (and resp. from Y) which have in G at least n/6 neighbours outside Y (resp. outside C). As |V_{1}| = 2n/3 + o(n) some x ∈ V_{2} satisfies d_{V_{2}}(x) ≥ n. But x necessarily belongs to Y ∪ C, say x ∈ C, and can have at most |Y ∩ V_{2}| + n/6 ≤ 5n/6 + o(n) V_{2}-neighbours, which is a contradiction proving c = o(n).

Using the above lower bounds on e_{a}, e_{c}, e_{y} and the inequality a ≥ n − c + 1 we obtain

e(G) = n^{2} + c_{g}n^{3/2} ≥ e_{a} + e_{c} + e_{y} − (n − a + 1)c
≥ n^{2} + n − \frac{c^{2}+3c}{2} + ac ≥ n^{2} + n + \frac{−3c^{2}+c(2n−1)}{2}.

Solving this (quadratic in c) inequality we obtain that necessarily c < c_{g}n^{1/2} for large n as c cannot be bigger than the larger root 2n/3 + o(n).

Writing e(G) ≥ e_{a} + e_{c} + e_{y} − (n − a + 1)c + e_{r} and substituting a ≥ n − c + 1 everywhere (as the total coefficient of a is positive) we obtain

c_{g}n^{3/2} ≥ \frac{−r^{2}+r(2n+1)}{2} + cn − 2cr + O(n).

The larger root of this quadratic in r inequality is 2n+o(n), but r ≤ n+o(n) since a = n + o(n) and a + r ≤ |L|. So we conclude that l − 1 = c + r ≤ c_{g}n^{1/2} + O(1) as required.

Now let us try to derive a final contradiction.

Proof of (105). Let x_{max} be a vertex of maximal degree Δ(G) = c_{m}n^{3/2}. The greedy algorithm shows that e(G) ≥ n^{2} + Δ(G), that is, c_{m} ≤ c_{g}. Let c′ = ((4 + c_{g}^{2})^{1/2} − c_{g})/2 and c_{f} = 1.732.

We apply a version of the greedy algorithm. Set initially A = C = ∅ and B = V (G).

At Stage 1 move to A, one by one and as long as possible, a vertex x ∈ B with d_{B\L}(x) ≥ n − l and d_{B∪C}(x) ≥ n. After x was moved do the n-check, that is, move to C all vertices y ∈ B ∩ L with d_{B∪C}(y) < n. We may assume that we were selecting x ∈ B so that d_{G}(x) was non-increasing. Let A_{1} be the set of vertices moved to A at Stage 1, F = {x ∈ A_{1} : d_{G}(x) ≥ n + c_{f}n^{1/2}} and a_{f} = |F |/n. By Lemma 81 we have l ≤ c_{g}n^{1/2} + o(1), so the number of edges incident to F is at least

∑_{x∈F} d(x) − ∑_{x∈F} \frac{d(x) − n + l}{2} ≥ a_{f}n^{2} + a_{f} \frac{c_{f} − c_{g}}{2} n^{3/2} + o(n^{3/2}).

At Stage 2 move to A, one by one and as long as possible, any vertex x ∈ B having at least n + c′n^{1/2} neighbours in B ∪ C and for every such x do the n-check as in Stage 1.

At Stage 3 we repeat the following until B ∩ L = ∅. Take x ∈ B ∩ L. As long as d_{B∪C}(x) ≥ n move to A some x-neighbour y ∈ B ∩L (note that d_{B∪C}(y) ≥ n) and perform the n-check. Such y necessarily exists as x has fewer than n − l neighbours in B \ L while |C| ≤ l. (The latter inequality is true because if |C| > l at some moment then continuing with the standard greedy algorithm applied to B ∪ C we find at least n − |A| + 1 vertices in B ∩ L which contradicts |L| = n + l.) Of course, the last n-check moves x itself to C.

Let a_{i}n (resp. c_{i}n^{1/2}) be the number of vertices moved to A (resp. to C) at the ith Stage. As eventually Δ(G[B ∪C]) < n we conclude that a_{1} +a_{2} +a_{3} > 1. Also a_{3} ≤ c′c_{3} as for every x moved to C at Stage 3 we moved at most c′n^{1/2} vertices to A.

Note that the first vertex moved at Stage 1 may be assumed to have degree Δ(G) = c_{m}n^{3/2} unless Δ(G) = O(n). So our algorithm produces the following lower bound on the size of G:

e(G) ≥ n^{2} + (c_{m} + a_{f} \frac{c_{f} −c_{g}}{2} + a_{2}c′ + c_{3}(1 − a_{3}) + o(1)) n^{3/2}.  (109)

Now using the inequalities a_{3} ≤ c′c_{3} (twice) and 0 ≤ c_{3} ≤ c_{g} + o(1) (by Lemma 81 we have c_{3} ≤ |C|n^{−1/2} ≤ c_{g} + o(1)) we obtain from (109) that

a_{2} + a_{3} ≤ \frac{c_{g} − a_{f}(c_{f} − c_{g})/2 − c_{m} − c_{3} + c′c_{3}^{2}}{c′} + c′c_{3} + o(1)
≤ \frac{c_{g} − a_{f}(c_{f} − c_{g})/2 − c_{m}}{c′} + max (0, c_{g}^{2} + c′c_{g} − c_{g}/c′) + o(1).

But our c′ satisfies c_{g}^{2} + c′c_{g} = c_{g}/c′ so the second term disappears.

Choose a set Y ⊂ \overline{L} by placing each vertex of \overline{L} into Y independently with probability p = (c_{f} + 2ε)n^{−1/2}, where ε > 0 denotes a small constant.

The number of Y -neighbours of any x ∈ L has a binomial distribution with expectation at most pc_{m}n^{3/2} = (c_{f} + 2ε)c_{m}n. Hence the probability that say d_{Y} (x) > (c_{f}c_{m} + 3ε)n is exponentially small in n by Chernoff’s bounds [Che52].

Similarly, the expected value of d_{Y} (x) for x ∈ A_{1} is at least p(n − l) ≈ (c_{f} + 2ε)n^{1/2} and d_{Y} (x) < (c_{f} + ε)n^{1/2} with probability at most exp(−cn^{1/2}) for some constant c > 0.

Hence, there exists Y (in fact, almost every choice would do) such that d_{Y} (x) ≤ (c_{f}c_{m} + 3ε)n for every x ∈ L and d_{Y} (y) ≥ (c_{f} + ε)n^{1/2} for every y ∈ A_{1}.

Now consider the partition V (G) = V_{1} ∪ V_{2}, where V_{1} = (\overline{L} \ Y) ∪ (A_{1} \ F). Any x ∈ A_{1} \ F has at least (c_{f} + ε)n^{1/2} > d(x) − n neighbours in Y, so d_{V_{1}}(x) < n. But then d_{V_{2}}(x) ≥ n for some x ∈ L ∩ V_{2}. Hence,

n ≤ |V_{2} \ Y | + d_{Y} (x) ≤ n + l − |A_{1}| + |F | + (c_{f}c_{m} + 3ε)n,

or equivalently

a_{2} + a_{3} + a_{f} + c_{f}c_{m} ≥ 1 + error term,  (110)

where the error term can be made arbitrarily small by choosing the constant ε small.

Chopping off some terms in (109) we obtain that a_{f} lies between 0 and 2(c_{g} − c_{m})/(c_{f} − c_{g}) + o(1). Hence

a_{2} + a_{3} + a_{f} ≤ \frac{c_{g} − a_{f}(c_{f} − c_{g})/2 − c_{m}}{c′} + a_{f} + o(1)
≤ max (\frac{c_{g}−c_{m}}{c′} , 2\frac{c_{g}−c_{m}}{c_{f} −c_{g}} ) + o(1).  (111)

Using the values of c_{g} and c_{f} we obtain from (110) and (111) that necessarily

max(0.767 + 0.403 c_{m}, 0.9992 + 0.0004 c_{m}) ≥ 1 + o(1),

which cannot be satisfied for 0 ≤ c_{m} ≤ 0.577. Remark. The constant 0.577 can be improved, even with the present proof. For example, the optimal choice

c_{f} = min (\sqrt{4 + c_{g}^{2}}, c_{g} + \sqrt{2(c_{g} − c_{m})/c_{m}}) ,

should give (with extra algebraic work) c_{g} ≥ 0.591.

Also, after Stage 2 we could apply the algorithm of Lemma 81: we have identified at least (c_{m} + a_{2}(c′ − c_{g}) + a_{f} \frac{c_{f} −c_{g}}{2} )n^{3/2} ‘useless’ (from the point of view of Lemma 81) edges, which should bring down the bound on l there. We do not know how much gain this would have given (the calculations get rather messy) but we believe that we have reached a good compromise in the sense that the proof is not too long and the bound is not too bad.

22.3 Cycles of Consecutive Lengths

As we already mentioned, Erdős and Faudree [EF99] study minimum graphs G such that if G is a union of two graphs, one having maximal degree less than n, then the other contains all odd cycles C_{m} with 3 ≤ m ≤ n − 3. Here we show, that if we require cycles lengths from 3 to Θ(n), then we can present a construction with only (1 + ε)n^{2} edges for any fixed ε > 0.

In the proof below we introduce constants c_{1}, c_{2}, and so on. It should not be hard to check that we can always choose c_{i} (depending on c_{1}, …, c_{i−1}) satisfying all conditions set in the proof. We do not try to optimize the constants.

Theorem 82 For any fixed ε > 0, there is a graph G with at most (1 + ε)n^{2} edges such that if E(G) is coloured blue-red without a blue K_{1,n}, then we have red cycles of all lengths (even and odd) between 3 and cn for some c = c(ε) > 0 which does not depend on n.

Proof. Choose integers

m = \sqrt{n/2} + O(1),
k = (\sqrt{2} + c_{1})\sqrt{n} + O(1),
l = n + c_{1}n + O(1),
h = c_{1}\sqrt{n} + O(1).

Choose k-sets K_{1}, …, K_{m}, l-sets L_{1}, …, L_{m}, and an h-set H (all disjoint). Let G consist of all edges intersecting H and of all edges intersecting K_{i} and lying within K_{i} ∪ L_{i}, i ∈ [m], that is, G = K_{h} + mP_{k,l}. If c_{1} > 0 is small, then G has at most (1 + ε)n^{2} edges.

Consider any blue-red colouring of E(G) without a blue K_{1,n}. Let G′ ⊂ G be the red subgraph, let d′(x) be the red degree of x ∈ V (G), and so on.

Define the bipartite graph F with classes H and [m] as follows; x ∈ H is connected to i ∈ [m] if and only if x sends at least l +c_{2}\sqrt{n} red edges to K_{i} ∪L_{i}, c_{2} = c_{1}/2. Now, the inequality

(m − d_{F} (x))c_{2}\sqrt{n} + d_{F} (x)k ≥ mk − n + 1,

implies that each x ∈ H has d_{F} (x) ≥ c_{3}\sqrt{n} neighbours in F.

First, let us show how to find red cycles of all lengths up to c_{4}\sqrt{n}. Choose, any {x, i} ∈ E(F).

In G′, we have d′_{K_{i}}(x) ≥ c_{2}\sqrt{n} and each vertex in Γ′_{K_{i}}(x) has at least c_{1}n + o(n) neighbours in Γ′_{K_{i}∪L_{i}}(x). (Because the latter set has size n + c_{1}n + O(\sqrt{n}) while we do not have a blue K_{1,n} in G.) Thus we have Θ(n^{3/2}) red edges within Γ′_{K_{i}∪L_{i}}(x). By the theorem of Erdős and Gallai [EG59], we have a red path of length c_{4}\sqrt{n} there, which together with x creates red cycles of all lengths up to c_{4}\sqrt{n}.

Next, the graph F (which, in fact, has positive density) has a cycle of length 2t = Θ(\sqrt{n}) with 4t < c_{4}\sqrt{n} for large n; let it go through vertices x_{1}, i_{1}, , …, x_{t}, i_{t}, x_{t+1} = x_{1}, where x_{j} ∈ H for j ∈ [t].

To prove the theorem it is clearly enough to show that, for any j ∈ [t], we can find a red path connecting x_{j} and x_{j+1} through K_{i_{j}} ∪ L_{i_{j}} of any length between 2 and c_{5}\sqrt{n}, for some constant c_{5}.

Consider X = (Γ′(x_{j})∪Γ′(x_{j+1}))∩(K_{i_{j}} ∪L_{i_{j}}). Now, X∩K_{i_{j}} has at least c_{2}\sqrt{n} elements, each being incident to at least c_{1}n+O(\sqrt{n}) red edges. It is not hard to see that we can find a red cycle C within X of length at least c_{5}\sqrt{n} intersecting Γ′(x_{j}) ∩ Γ′(x_{j+1}). (The latter set has size n + c_{1}n + O(\sqrt{n}) and it is incident to almost all red edges lying within X.) It is easy to see that we can additionally require that C has a red cord E. Now, by a simple lemma (which is implicit in Bondy and Simonovits [BS74] and explicit in Verstraëte [Ver99]), C +E contain paths connecting Γ′(x_{j}) ∩ V (C) to Γ′(x_{j+1}) ∩ V (C) (two intersecting sets that cover V (C)) of all lengths from 0 to v(C) − 1, as required.

Remark. For each j ∈ [t], we can find a red cycle of any prescribed length between 3 and c_{4}\sqrt{n} lying within K_{i_{j}} ∪L_{i_{j}} ∪\{x_{j}\}. Hence, we can find t = Θ(\sqrt{n}) such vertex disjoint cycles in G. Of course, one can try to prove many other similar results about our graph G. For example, what is c = c(ε) if ε tends to zero with a given rate as n → ∞, say ε = Θ(1/\sqrt{n})? But we do not want to build a whole theory out of it: our purpose was to demonstrate that if we allow (1 + ε)r̂(K_{1,n}, C_{odd}) edges then we can witness much stronger properties.

23 Removing Vertices

In this section we denote l = k − j ≥ 0. Thus a graph of order n + k belongs to M(n, k, j) if after the removal of any l vertices the maximal degree is at least n.

23.1 Some Constructions

Here is our counterexample to the conjecture of Erdős, Reid, Schelp and Staton [ERSS96, Conjecture 1].

Example 83 The formula (106) is not true if n ≤ (j − 2)l.

Proof. Write n = lq + r with 0 ≤ r < l. Let A = [l + 1], y = l + 2, and R = [l +3, l +r +2], that is, R ⊂ X \(A∪{y}) is a set of size r. Our assumption on n implies that j ≥ q + 2, that is,

n + k − l − r − 2 ≥ (l + 1)q.

Therefore, in X \ (A ∪ R ∪ {y}) we can choose disjoint q-sets Q_{1}, …, Q_{l+1}. Let our graph G consist of the following edges: {f, h} ∈ A^{(2)} with |f − h| > 1 (that is, A spans the complete graph but for a Hamiltonian path), all edges between A and R, all edges connecting f ∈ A to Q_{h} with h ≠ f and edges {f, y}, f ∈ [2, l]. Thus all vertices in A have degree n + l − 1. It is easy to check that the size of G is by one smaller than the bound given by (106).

We claim that G ∈ M(n, k, j). Suppose on the contrary to our claim that we can remove some set L of size l so that the remaining graph has maximal degree less than n. Let x be any vertex in A \ L which is not empty as |A| > l. As the degree of x should be less than n now, we conclude that x is connected to each vertex in L. Therefore, any vertex in A non-incident to some x ∈ A \ L lies itself in A \ L. As \overline{G}[A] is connected (it is a path), we conclude that A ∩ L = ∅. But the set of vertices connected (in G) to everything in A is precisely R and it has r < l elements, which is a contradiction.

Remark. In Example 83 we can win a few more edges if n is yet smaller. As before, we let A = [l + 1] and y = l + 2. Suppose that for some p we can squeeze into [l + 3, n + k] an r-set R and q-element sets Q_{1}, …, Q_{p+1} with r < p and pq + r = n. To define G, let \overline{G}[A] consist of h = ⌊\frac{l+1}{p+1}⌋ vertex-disjoint paths of length p + 1 each; for every such path (x_{1}, …, x_{p+1}) we connect x_{i}, i ∈ [p + 1], to everything in R and in Q_{j}, j ≠ i, and we connect x_{i}, i ∈ [2, p], to y. Also, we add some extra edges so that any vertex of A not on a path has degree n + l.

Suppose that G ∉ M(n, k, j), that is, there is an l-set L with Δ(G − L) < n. Let x_{0} ∈ A \ L. It must lie on a path P. (Otherwise d_{G}(x) = n + l.) Like in Example 83 we argue that P does not intersect L and, in fact, every x ∈ P is connected to all vertices in L. But the number of vertices connected to the whole of P is |(L \ P) ∪ R| < l, which is a contradiction.

It is easy to see that we have h edges less than in (106), so it is advantageous to choose p as small as possible. The condition we have to satisfy is

n + k − l − 2 ≥ r + (p + 1)q

or, equivalently, j − 2 ≥ q = ⌊n/p⌋. Therefore, we choose p = ⌈\frac{n}{j−2}⌉. Note that our gain compared to (106) is h = ⌊\frac{l+1}{p+1}⌋.

Erdős et al [ERSS96] observe that (106) is not true ‘when k is very large compared to n.’ Here is an example, for any given l and n, giving only a fraction of (106) with k moderately small (starting with k ≥ n + l + 1).

Example 84 Suppose that n + k ≥ pn + l + 1, where p is an integer greater than 1. Take a representation l + 1 = l_{1} + … + l_{p} and let G = ⊔_{i∈[p]}(K_{l_{i}} + E_{n}). Then G ∈ M(n, k, k − l).

Proof. Let L ⊂ V (G) be any l-set. There must exist i ∈ [p] such that L intersects the corresponding component C_{i} in less then l_{i} vertices. Hence, at least one vertex in K_{l_{i}} survives and it has at least n neighbours outside L.

23.2 Improving Condition (107)

We can prove the following (which is an improvement of (107) if j ⪅ k/3).

Theorem 85 Let j ≥ 2 and n ≥ 14. Then (106) is true if

n ≥ (j + \frac{1}{2}) l + \frac{2j + l}{4j − 2}.

Proof. Let G ∈ M(n, k, j). To prove the theorem by induction is it enough to show that maximal degree of G is at least n + l. (Because removing a vertex from G we obtain a graph in M(n, k − 1, j) and clearly m(n, j, j) = n.)

Let H = {x ∈ V (G) : d(x) ≥ n} and h = |H|. Let us show that h is not large by applying the following procedure to G.

Let A = C = ∅ and B = V (G). Repeat the following as long as possible or until |A| = l + 1. Move to A any vertex x ∈ B (if it exists) having at least n neighbours in B. For every such x do the n-check, that is, move to C all y ∈ B∩H with d_{B∪C}(y) < n. (In fact, for every such y we have d_{B∪C}(y) = n−1.)

Suppose we have stopped. Let a, b, c be the sizes of the eventual sets A, B, C. Inductively, we find a set Y = {y_{1}, …, y_{l+1−a}} ⊂ B ∪ C such that each y_{i} has at least n neighbours in C ∪ B \ {y_{1}, …, y_{i−1}}. As each y ∈ C has fewer than n neighbours in B ∪ C, we conclude that Y ⊂ B. Let R = (B \ Y) ∩ H and r = |C ∪ R|. Each x ∈ R has at least n neighbours in C ∪ B for otherwise it would belong to C.

Counting the number of edges encountered in our algorithm we obtain that

e(G) ≥ an + |Y |n + r(n − 1) − \binom{r}{2} − r|Y |.

Using a + |Y | = l + 1 (and the trivial inequality |Y | ≤ l) we obtain

\binom{l+1}{2} ≥ r (n − 1 − \frac{r−1}{2} − |Y |) ≥ r (n − l − \frac{r+1}{2}) .

To satisfy this quadratic in r inequality, r must not lie between the roots r_{1,2} = n − l − \frac{1}{2} ± R, where

R = \frac{1}{2}\sqrt{4n^{2} − 4n(2l + 1) + 1}.

The assumption of the theorem implies that

l ≤ 3n/8.  (112)

Using (112), one can check that R ≥ (n − 3)/2. Suppose that r ≥ r_{2}. Observe that

r_{2} ≥ n − l − \frac{1}{2} + \frac{n−3}{2} = \frac{3}{2}n − l − 2 ≥ \frac{9}{8}n − 2.  (113)

As before, the inequality e(G) ≥ an + |Y |n + r \frac{n−1}{2} − r|Y | implies that

\binom{l+1}{2} ≥ r(\frac{n−1}{2} − l).  (114)

Using (112) and (113), we can deduce from (114) that n ≥ \frac{9}{128}n^{2} + 1, which cannot be satisfied for n ≥ 14.

The above contradiction implies that r ≤ r_{1}; then we have

h = r + l + 1 ≤ n + \frac{1}{2} − R.  (115)

Suppose on the contrary that k > j and Δ(G) < n + l. For every x ∈ H we choose a j-set D_{x} ⊂ \overline{Γ(x)} and let D = ∪_{x∈H}D_{x}. We have |D| ≤ jh and we claim that this does not exceed n + j. To verify this, it is enough to check by (115) that jR ≥ jn − n − j/2. Squaring, we obtain that the latter is equivalent to n(2j − 1) ≥ 2j^{2}l + j, which is precisely our assumption.

Complete D to an arbitrary (n+j)-set E. As G ∈ M(n, k, j), some x ∈ E∩H has at least n neighbours in E, which is a contradiction as, by definition, E contains at least j non-neighbours of x.

Hence, Δ(G) ≥ n + l and the theorem follows by induction.

24 Splitting into Parts

Here we consider b(n, m) = min{e(G) : G ∈ B(n, m)}, where B(n, m) consists of all graphs G such that, for any partition A ∪ B = V (G), either Δ(G[A]) ≥ n or Δ(G[B]) ≥ m.

Clearly, b(n, m) = b(m, n). Let us assume n ≥ m.

24.1 General Bounds

The following simple argument gives a very good general lower bound on b(n, m).

Let G ∈ B(n, m) be any graph. Set initially A = V (G) and B = ∅. As long as |B| ≤ m, move to B any x ∈ A with d_{A}(x) ≥ n. (Such a vertex exists, because obviously Δ(G[B]) < m.)

When we finish, |B| = m + 1. Swap the sets A and B each with the other. (So that now |A| = m + 1.) Next, consecutively and as long as possible, move to A any vertex of G[B] of degree at least m. As eventually Δ(G[B]) < m, our assumption on G implies that |A| ≥ n + 1 (to allow a vertex of degree at least n). Counting the edges encountered in this procedure, we obtain the following bound valid for any n and m.

b(n, m) ≥ (m + 1)n + ((n + 1) − (m + 1))m = 2mn − m^{2} + n.  (116)

Next, we provide a general construction giving an upper bound on b(n, m).

Example 86 Choose representations m = m_{1} + … + m_{f} and n − m = n_{1} + … + n_{g}. Let G be the disjoint union of P_{m_{i},n}, i ∈ [f], and P_{n_{j},m}, j ∈ [g], plus a vertex x connected to everything else. We claim that G ∈ B(n, m).

Proof. Let V (G) = A ∪ B be any partition. Case 1 Suppose x ∈ A. Observe that at least m_{i} vertices from each P_{m_{i},n} and at least n_{j} vertices from each P_{n_{j},m} lie in A. (Otherwise Δ(G[B]) ≥ m.) But then

d_{A}(x) = |A| − 1 ≥ ∑_{i∈[f]} m_{i} + ∑_{j∈[g]} n_{j} = n.

Case 2 If x ∈ B (and Δ(G[A]) < n), then from each P_{m_{i},n} at least m_{i} vertices go to B and d_{B}(x) ≥ ∑_{i∈[f]} m_{i} = m, as required.

Let us compute how many edges we use in Example 86.

b(n, m) ≤ e(G) = n+fn+gm+mn+(n−m)m+ ∑_{i∈[f]} \binom{m_{i}}{2}+ ∑_{j∈[g]} \binom{n_{j}}{2}, (117)

To minimize it, we let the m_{i}’s (and the n_{j}’s) be nearly equal while f and g have to be around m(2n)^{−1/2} and (n−m)(2m)^{−1/2} respectively. Putting bounds (116) and (117) together we obtain the equality (108) claimed in the introduction.

24.2 Small Fixed m

In the extreme case when m is fixed and n tends to infinity, we consider Example 86 with f = 1 (so m_{1} = m) and g = n(2m)^{−1/2} + O(1). Then

∑_{j∈[g]} \binom{n_{j}}{2} < g \frac{(\frac{n}{g} + 1) \frac{n}{g}}{2} ≤ \frac{n}{2}(\sqrt{2m} + 1) + O(1),

and we obtain the following.

Corollary 87 Let m ≥ 1 be fixed. Then b(n, m), n ∈ ℕ, lie between two linear functions, namely

(2m + 1)n + O(1) ≤ b(n, m) ≤ (2m + \sqrt{2m} + \frac{5}{2})n + O(1).

However, for a few particular small instances of m we can be more precise.

Let us provide a construction of G ∈ B(n, 1), n ≥ 2. Represent n = 2k+l+1 and let G be disjoint union of k triangles, l disjoint edges, plus vertices x, y; x is connected to every other vertex while y is connected to some n vertices (besides x). Clearly, e(G) = 3k + 3k + l + 2l + n + 1 = 4n − 2.

To show that G ∈ B(n, 1), suppose that we have a partition V (G) = A ∪ B with B being an independent set. If one of x or y belongs to B, then A contains the other plus their n common neighbours and so Δ(G[A]) ≥ n. If {x, y} ⊂ A, then at least 2 vertices from each triangle and at least 1 vertex from each edge must be in A and d_{A}(x) ≥ 1 + 2k + l = n, as required.

Theorem 88 For any n ≥ 8, b(n, 1) = 4n−2 and all extremal graphs are given by the above construction.

Proof. Let n ≥ 1 and let G be any graph in B(n, 1) of size at most 4n − 2. Let L be the set of vertices of G of degree at least n. Clearly, |L| > 1.

First, suppose that |L| = 3, say L = {x, y, z}. The partition with B = L shows that L is not independent in G. Case 1 G[L] consists of a single edge, say {x, y}. The partition with B = {x, z} (resp. B = {y, z})) shows that y (resp. x) has at least n neighbours outside L. Hence, L is incident to at least 3n + 1 edges, and we have at most (4n − 2) − (3n + 1) = n − 3 edges non-incident to L. Letting A consist of all vertices of L plus an arbitrary endvertex of each edge outside L, we obtain a contradiction: \overline{A} is independent while |A| ≤ n. Case 2 G[L] consists of two edges, say {x, y} and {x, z}. The partition with B = \{y, z\} shows d\overline{L}(x) ≥ n. The partition with B = \{x\} shows that another vertex of L has at least n neighbours outside L. Hence, L is incident to at least 3n + 1 edges, and we can derive a contradiction as above.

Case 3 G[L] is the complete graph. Placing in B a vertex of L, we deduce that some two vertices in L, say x and y, have at least n + 1 neighbours each. Thus, we have already found 3n − 1 edges incident to L; so we can have at most one more such edge. Case 3.1 Suppose that d(z) = n and d(x) ≥ d(y). (That is, d(y) = n + 1.) Every neighbour u of y is connected to x. (Otherwise consider B = {x, u}.) This means that |U| = n−1, where U = (Γ(x)∪Γ(y))\L. Choose any u ∈ U\Γ(z) ≠ ∅. The partition with B = {z, u} shows that d(x) = n + 2. By letting A = L and consecutively moving to A a non-isolated vertex of G[\overline{A}], we conclude that G[\overline{L}] consists of n − 2 disjoint edges. (And e(G) = 4n − 2.) Furthermore, if n ≥ 5, we can choose an independent 3-set C ⊂ Γ\overline{L}(x). If z sends at least one edge to C, let B = C; otherwise let B = C ∪ {z}. It is easy to see that in either case Δ(G[\overline{B}]) < n, which is a contradiction. Case 3.2 Suppose d(x) = d(y) = d(z) = n + 1. As before, we conclude that G[\overline{L}] consists of n − 2 disjoint edges. (And e(G) = 4n − 2.) Let n ≥ 7. Clearly, |V | ≤ 2, where V = Γ(x) ∩ Γ(y) ∩ Γ(z). (Otherwise, consider any independent 2-set B ⊂ L.) Also, there is no v ∈ (Γ(x) ∩ Γ(y)) \ Γ(z). (Otherwise, let B = \{v, z\}.) But then, for n ≥ 5, we can choose non-incident u, v ∈ \overline{Γ(z)} with u ∈ Γ(x) and v ∈ Γ(y), and the consideration of B = {u, v, z} yields a contradiction.

Similarly, but with less effort, we can exclude the case |L| ≥ 4 for n ≥ 6. So, we conclude that L = {x, y}. Considering the partition with B = {x, y}, we see that x is connected to y. Considering the partition with B = {x} or B = {y}, we conclude that d(x) ≥ n + 1 and d(y) ≥ n + 1.

We apply the following procedure. Let A consist of x and y plus all vertices connected neither to x nor to y; let B = \overline{A}. At Stage 0 consecutively move to A a vertex of degree at least 3 in G[B]. Stage 1: one by one and as long as possible, move to A a vertex of degree 2 in G[B].

Now, G[B] consists of isolated edges. Stage 2: for each edge {a, b} ∈ E(G[B]) with Γ(a) ∩ L ⊂ Γ(b) ∩ L we move a to A but keep b in B.

After this stage each edge in G[B] together with L spans a C_{4}; let s_{3} = e(G[B]). For j ∈ [0, 2] let s_{j,i} be the number of vertices moved to A at Stage j which were incident to i vertices in L, i ∈ [2], and let s_{j} = s_{j,1} + s_{j,2}.

Case 4 Suppose that Δ(G[A]) ≥ n; let d_{A}(x) ≥ n. Then the total number of edges in G is at least n+1 (edges incident to y) plus 3(n−1). (Because for each of n − 1 vertices incident to x which were moved to A we count at least three edges; for example, for a vertex a moved at Stage 2, we encounter the edges {a, x}, {a, b} ∈ E(G[B]) and {b, x}.) Hence, e(G) ≥ 4n − 2 as required. Case 5 Suppose that Δ(G[A]) < n. As we can make B ⊂ V (G) independent by moving an arbitrary endvertex of each edge to A (and after this we must have Δ(G[A]) ≥ n), we conclude that now

2n − 1 ≤ s_{3} + d_{A}(x) + d_{A}(y) = s_{3} + 2 + ∑_{j=0}^{2}(s_{j,1} + 2s_{j,2}).  (118)

On the other hand, we have the following estimate.

e(G) ≥ 4s_{0,1} + 5s_{0,2} + 3s_{1,1} + 4s_{1,2} + max(s_{1} − s_{2} − s_{3}, 0) + 3s_{2,1} + 5s_{2,2} + 3s_{3} + 1.

(119)

Only the max-term needs some explanation. After Stage 1 G[B] consists of s_{3} + s_{4} isolated edges. Let us move back to B the s_{1} vertices moved at Stage 1. As the resulting graph has maximal degree 2, we must use at least s_{1}−s_{2}−s_{3} new vertices. Each of these vertices sends at least one edge to L, which constitutes the extra term. If we multiply (118) by 2 and substitute this from (119), we obtain (using e(G) ≤ 4n − 2)

s_{3} + 2s_{0,1} + s_{0,2} + s_{1,1} + s_{2} + max(s_{1} − s_{2} − s_{3}, 0) ≤ 3.

Hence, s_{1} ≤ 3 (and s_{2} + s_{3} ≤ 3). From (118) we deduce that

n ≤ (s_{1} + s_{2}) + (s_{3} + 3)/2 ≤ 7\frac{1}{2}.

Hence, we have shown that b(n, 1) = 4n−2, for n ≥ 8. Conversely, if a graph G achieves this bound, then |L| = 2, s_{0} = 0, s_{3} = 0, one vertex of L is connected to every \overline{L}-neighbour of the other L-vertex, and every vertex moved at Stage 1 belonged to an isolated triangle of G[B]. Now the required characterization follows. The details are left to the reader.

Remark. Perhaps, b(n, 1) = 4n−2 for any n ≥ 2 (clearly b(1, 1) = 3), but then there are many other constructions achieving this bound. A direct search is feasible (note that our proof of Theorem 88 contains some information reducing this search), but it would be too long to include here.

Theorem 89 b(n, 2) = 6n + O(1).

Proof. A construction of G ∈ B(n, 2) first. Consider ⌈\frac{n}{2}⌉ disjoint 4-cycles and one triangle, say on X = {x_{1}, x_{2}, x_{3}}. To this we add some further edges: x_{1} is connected to every other vertex while x_{2} and x_{3} are connected to some fixed m-set C ⊂ Γ(x_{1}) \ {x_{2}, x_{3}}.

Let V (G) = A ∪ B be any partition with Δ(G[B]) ≤ 1. If x_{1} ∈ A, then at least 2 vertices of each C_{4} belong to A and d_{A}(x_{1}) = |A| − 1 ≥ n, as required. If x_{1} ∈ B, then all but at most one vertex in C ∪ {x_{2}, x_{3}} lie in A and a vertex in X ∩ A ≠ ∅ has at least |C| = n neighbours in A. Hence, G ∈ B(n, 2) and b(n, 2) ≤ 6n + O(1), as required.

We show the lower bound. Let G ∈ B(n, 2) be any graph with at most 6n edges; we have to deduce 6n − e(G) = O(1). Let L = {x ∈ V (G) : d(x) ≥ n}.

If |L| ≥ 4 then we have at least 4n + O(1) edges incident to L. Let A = L and B = \overline{A}. As long as possible, move to A a vertex of G[B] of degree at least

2. Before we stop, we repeat the iteration at least n + 1 − |L| = n + O(1) times, which means that there are 2n + O(1) edges not incident to L and we are home.

Clearly, |L| ≥ 2. (Otherwise the partition with B = L contradicts G ∈ B(n, 2).) Hence, |L| = 3 and the theorem follows from Lemma 90 below.

The following related notion is useful. Let B′(n, m, l) be the class of graphs G with a fixed l-set L ⊂ V (G) such that d(x) ≥ n, x ∈ L, and for any partition V (G) = A ∪ B with Δ(G[B]) < m and L ⊂ A some vertex x ∈ L has at least n neighbours in A. Also, denote b′(n, m, l) = min{e(G) : G ∈ B′(n, m, l)}.

Lemma 90 For l ∈ [3], b′(n, 2, l) ≥ (3 + l)n + O(1).

Proof. Let G ∈ B′(n, 2, l) be any graph. We may freely remove any vertex incident to no vertex of the selected set L = {x_{1}, …, x_{l}}, as this does not violates the B′(n, 2, l)-property. Let

Γ_{A} = \{y ∈ \overline{L} : \{y, x_{i}\} ∈ E(G) iff i ∈ A\} , A ⊂ [l].

Case 1 Let l = 1. Let A = L and B = V (G) \ L. As long as possible, move to A any y ∈ B with d_{B}(y) ≥ 3. At the end, G[B] consists of disjoint cycles, paths and vertices. But we can move to A at most ⌊\frac{p+2}{3}⌋ (resp. ⌊\frac{p+1}{3}⌋) vertices from each cycle (resp. path) of length p to ensure Δ(G[B]) < 2. As the number of moved vertices must be at least n and we use at least 4 edges per vertex (including edges incident to x_{1}), the claim follows. Case 2 Let l = 2. We apply an inductive on n argument, ensuring that we have at least 5 edges per every removed vertex, except in O(1) cases. First, whenever we have y ∈ \overline{L} with d_{G}(y) ≥ 5, we remove it, obtaining a graph in B′(n−1, 2, 2). Next, if we have y_{1} ∈ Γ_{1} and y_{2} ∈ Γ_{2} at distance at least 3, we contract them without loosing the B′(n, 2, 2)-property. Suppose we are finally stuck and suppose |Γ_{1}| ≤ |Γ_{2}|. As Δ(G[\overline{L}]) < 4, we conclude that g = |Γ_{1}| = O(1). Removing Γ_{1} from G, we obtain a graph in B′(n − g, 2, 2); further, removing x_{2} (and at least n − g edges) we obtain a graph in B′(n − g − 1, 2, 1) which has size at least 4n + O(1). Hence, b′(n, 2, 2) ≥ 5n + O(1). Case 3 Let m = 3. Like in Case 2, we remove a vertex x ∈ \overline{L} of degree at least 6; also, we contract any y ∈ Γ_{A}, z ∈ Γ_{B}, at distance at least 3 for A ∩ B = ∅. Next, removing O(1) vertices we ensure that all but one of Γ_{i}, i ∈ [3], are empty, say Γ_{1} = Γ_{2} = ∅. Also, we make either Γ_{12} or Γ_{3} empty. If Γ_{12} = ∅, then Γ(x_{1}) ⊂ Γ(x_{3}); removing x_{1} (and ≥ n + O(1) edges) we obtain a graph in B′(n + O(1), 2, 2) of size at least 5n + O(1)—we are home.

So, suppose Γ_{3} = ∅. If possible, remove any three vertices in respectively Γ_{12}, Γ_{13}, Γ_{23} incident to at least 12 edges to obtain a graph in B′(n − 2, 2, 3). Removing up to O(1) vertices, we can assume that d\overline{L}(y) = 1 for each y in, for example, Γ_{12}. Let z ∈ \overline{L} be the neighbour of some y ∈ Γ_{12}. If d\overline{L}(z) = 1, then we can remove y, z from G without violating the B′(n, 2, 3)-property; otherwise, removing y, z we remove at least 6 edges and obtain a B′(n − 1, 2, 3)-graph. Eventually, we achieve Γ_{12} = ∅, that is, Γ(x_{1}) ⊂ Γ(x_{3}) and we are home again by Case 2.

Remark. In the next case m = 3 we can only show that

7n + O(1) ≤ b(n, 3) ≤ 9n + O(1).

References

[AEHK96] N. Alon, P. Erdős, R. Holzman, and M. Krivelevich. On k-saturated graphs with restrictions on the degrees. J. Graph Theory, 23:1–20, 1996.

[AF86] R.P. Anstee and Z. Füredi. Forbidden submatrices. Discrete Math., 62:225–243, 1986.

[AGS97] R. P. Anstee, J. R. Griggs, and A. Sali. Small forbidden configurations. Graphs Combin., 13:97–118, 1997.

[Alo85] N. Alon. An extremal problem for sets with applications to graph theory. J. Combin. Theory (A), 40:82–89, 1985.

[And67] I. Anderson. Some Problems in Combinatorial Number Theory. PhD thesis, Univ. of Nottingham, 1967.

[And87] I. Anderson. Combinatorics of Finite Sets. Oxford Univ. Press, 1987.

[Ans95] R. P. Anstee. Forbidden configurations: Induction and linear algebra. Europ. J. Combin., 16:427–438, 1995.

[BCE^{+}96] C. A. Barefoot, L. H. Clark, R. C. Entringer, T. D. Porter, L. A. Székely, and Z. Tuza. Cycle-saturated graphs of minimum size. Discrete Math., 150:31–48, 1996.

[Bei68] L. W. Beineke. On derived graphs and digraphs. In Beitr. Graphentheorie, Int. Kolloquium Manebach (DDR), pages 17–23, 1968.

[Blo87] A. Blokhuis. More on maximal intersecting families of finite sets. J. Combin. Theory (A), 44:299–303, 1987.

[BMRR80] L. D. Baumert, R. J. McEliece, R. J. Rodermich, and H. Rumsey. A probabilistic version of Sperner’s theorem. Ars Combinatoria, 9:91– 100, 1980.

[Bol65] B. Bollobás. On generalized graphs. Acta Math. Acad. Sci. Hung., 16:447–452, 1965.

[Bol67a] B. Bollobás. Determination of extremal graphs by using weights. Wiss. Z. Hochsch. Ilmenau, 13:419–421, 1967.

[Bol67b] B. Bollobás. On a conjecture of Erdős, Hajnal and Moon. Amer. Math. Monthly, 74:178–179, 1967.

[Bol67c] B. Bollobás. Weakly k-saturated graphs. In Proc. Coll. Graph Theory, pages 25–31. Ilmenau, 1967.

[Bol74] B. Bollobás. Three-graphs without two triples whose symmetric difference is contained in a third. Discrete Math., 8:21–24, 1974.

[Bol76] B. Bollobás. On complete subgraphs of different orders. Math. Proc.

Camb. Phil. Soc., 79:19–24, 1976.

[Bol78] B. Bollobás. Extremal Graph Theory. Academic Press, London, 1978.

[Bol86] B. Bollobás. Combinatorics, Set Systems, Families of Vectors, and Combinatorial Probability. Cambridge Univ. Press, 1986.

[Bol95] B. Bollobás. Extremal graph theory. In R. L. Graham, M. Grötschel, and L. Lovász, editors, Handbook of Combinatorics, pages 1231–1292. Elsevier Science B.V., 1995.

[Bon72a] J. A. Bondy. Induced subsets. J. Combin. Theory (B), 12:201–202, 1972.

[Bon72b] J. A. Bondy. Variations on the hamiltonian scheme. Can. Math. Bull., 15:57–62, 1972.

[Bou74] N. Bourbaki. Algebra I. Reading: Addison-Wesley, 1974.

[BP69] L. W. Beineke and R. E. Pippert. The number of labelled k-dimensional trees. J. Combin. Theory, 6:200–205, 1969.

[BP77] L. W. Beineke and R. E. Pippert. On the structure of (m, n)-trees.

In Proc. 8th Southeast Conference on Combinatorics, Graph Th. and Computing (Louisiana State Univ., Baton Rouge), volume XIX of Congressus Numeratium, pages 75–80. Utilitas Math., Winnipeg, 1977.

[BS74] J. A. Bondy and M. Simonovits. Cycles of even length in graphs. J.

Combin. Theory (B), 16:97–105, 1974.

[BTK51] N. G. de Bruijn, C. Tengbergen, and D. Kruyswijk. On the set of divisors of a number. Nieuw Arch. Wiskd., 23:191–193, 1951.

[Cam95] P. J. Cameron. Counting two-graphs related to trees. Electronic J.

Combin., 2(#R4):8pp, 1995.

[Cay89] A. Cayley. A theorem on trees. Quart. J. Math., 23:376–378, 1889.

[CCES86] L. H. Clark, R. P. Crane, R. C. Entringer, and H. D. Shapiro. On smallest maximally non-hamiltonian graphs. In Combinatorics, Graph Theory and Computing, Proc. 17th Southeast. Conf., Boca Raton, volume 53 of Congres. Numer., pages 215–220, 1986.

[CE83] L. H. Clark and R. C. Entringer. Smallest maximally non-hamiltonian graphs. Period. Math. Hungar, 14:57–68, 1983.

[CES92] L. H. Clark, R. C. Entringer, and H. D. Shapiro. Smallest maximally nonhamiltonian graphs II. Graphs Combin., 8:225–231, 1992.

[Che52] H. Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statistics, 23:493–507, [Che93] W. Y. C. Chen. A coding algorithm for Rényi trees. J. Combin. Theory

(A), 63:11–25, 1993.

[Chu97] F. R. K. Chung. Open problems of Paul Erdős in graph theory. J.

Graph Theory, 25:3–36, 1997.

[CL85] C.-Y. Chao and N.-Z. Li. Trees of polygons. Archiv Math., 45:180–185, 1985.

[CO82] F. Chin and G. Ozsoyoglu. Auditing and inference control in statistical databases. IEEE Trans. Software Eng., SE–8:574–582, 1982.

[dCKW88] D. de Caen, D. L. Kreher, and J. Wiseman. On constructive upper bounds for the Turán numbers T (n, 2r + 1, 2r). In Combinatorics, Graph Theory and Computing, Proc. 19th Southeast. Conf., Baton Rouge, volume 65 of Congres. Numer., pages 277–280, 1988.

[DDFL85] S. J. Dow, D. A. Drake, Z. Füredi, and J. A. Larson. A lower bound for the cardinality of a maximal family of mutually intersecting sets of equal size. In Combinatorics, Graph Theory and Computing, Proc. 16th Southeast. Conf., Boca Raton, volume 48 of Congres. Numer., pages 47–48, 1985.

[Dew74] A. K. Dewdney. Multidimensional tree-like structures. J. Combin.

Theory (B), 17:160–169, 1974.

[DH56] G. B. Dantzig and A. J. Hoffman. Dilworth’s theorem on partially ordered sets. In H. W. Kuhn and A. W. Tucker, editors, Linear Inequalities and Related Systems, volume 38 of Annals Math. Studies, pages 207–214. Princeton Univ. Press, 1956.

[DH86] D. A. Duffus and D. Hanson. Minimal k-saturated and color critical graphs of prescribed minimum degree. J. Graph Theory, 10:55–67, 1986.

[Dil50] R. P. Dilworth. A decomposition theorem for partially ordered sets.

Annals of Math., 51(2):161–166, 1950.

[Doo73] M. Doob. An interrelation between line graphs, eigenvalues, and matroids. J. Combin. Theory (B), 15:40–50, 1973.

[EF99] P. Erdős and R. Faudree. Restricted size Ramsey numbers for cycles and stars. To appear in Discrete Math., 1999.

[EFT91] P. Erdős, Z. Füredi, and Z. Tuza. Saturated r-uniform hypergraphs.

Discrete Math., 98:95–104, 1991.

[EG59] P. Erdős and T. Gallai. On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hung., 10:337–356, 1959.

[EG61] P. Erdős and T. Gallai. On the minimal number of vertices representing the edges of a graph. Publ. Math. Inst. Hungar. Acad. Sci., 6:181–203, [EH94] P. Erdős and R. Holzman. On maximal triangle-free graphs. J. Graph

Theory, 18:585–594, 1994.

[EHM64] P. Erdős, A. Hajnal, and J. W. Moon. A problem in graph theory.

Amer. Math. Monthly, 71:1107–1110, 1964.

[Eng97] K. Engel. Sperner Theory. Cambridge Univ. Press, 1997.

[ER66] J. Edmonds and G.-C. Rota. Submodular set functions (Abstract).

Waterloo Combinatorics Conference, 1966.

[Erd81] P. Erdős. Problems and results in graph theory. In G. Chartrand, editor, The Theory and Applications of Graphs, pages 331–341. John Wiley, New York, 1981.

[Erd99] P. Erdős. A selection of problems and results in combinatorics. Combin.

Prob. Computing, 8:1–6, 1999.

[ERSS96] P. Erdős, T. J. Reid, R. Schelp, and W. Staton. Sizes of graphs with induced subgraphs of large maximum degree. Discrete Math., 158:283– 286, 1996.

[ES83] P. Erdős and M. Simonovits. Supersaturated graphs and hypergraphs.

Combinatorica, 3:181–192, 1983.

[ES88] O. Egecioglu and L.-P. Shen. A bijective proof for the number of labelled q-trees. Ars Combinatoria, 25:3–30, 1988.

[FFP87] P. Frankl, Z. Füredi, and J. Pach. Bounding one-way differences.

Graphs Combin., 3:341–347, 1987.

[FHPZ98] Z. Füredi, P. Horak, C. M. Pareek, and X. Zhu. Minimal oriented graphs of diameter 2. Graphs Combin., 14:345–350, 1998.

[Foa71] D. Foata. Enumerating k-trees. Discrete Math., 1:181–186, 1971.

[Fra82] P. Frankl. An extremal problem for two families of sets. Europ. J.

Combin., 3:125–127, 1982.

[Fra95] P. Frankl. Extremal set systems. In R. L. Graham, M. Grötschel, and L. Lovász, editors, Handbook of Combinatorics, pages 1293–1330. Elsevier Science B.V., 1995.

[FRS97] R. J. Faudree, C. C. Rousseau, and R. H. Schelp. Problems in graph theory from Memphis. In R. L. Graham, editor, The Mathematics of Paul Erdős, volume 2, pages 7–26. Springer, Berlin, 1997.

[FS94] Z. Füredi and A. Seress. Maximal triangle-free graphs with restrictions on degrees. J. Graph Theory, 18:11–24, 1994.

[Gho75] S. P. Ghosh. Consecutive storage of relevant records with redundancy.

Comm. ACM, 18:464–471, 1975.

[GI75] C. Greene and G. A. Iba. Caley’s formula for multidimensional trees.

Discrete Math., 14:1–11, 1975.

[GK76] C. Greene and D. J. Kleitman. Strong versions of Sperner’s theorem.

J. Combin. Theory (A), 20:80–88, 1976.

[GKP89] R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete Mathematics:

a Foundation for Computer Science. Addison-Wesley Publ. Comp., 1989.

[GKP95] D. M. Gordon, G. Kuperberg, and O. Patashnik. New constructions for covering designs. J. Combin. Designs, 3:269–284, 1995.

[GPKS96] D. M. Gordon, O. Patashnik, G. Kuperberg, and J. H. Spencer. Asymptotically optimal covering designs. J. Combin. Theory (A), 75:270–280, 1996.

[Gri77] J. R. Griggs. Sufficient conditions for a symmetric chain order. SIAM J. Appl. Math., 32:807–809, 1977.

[GST98] G. R. Griggs, M. Simonovits, and G. R. Thomas. Extremal graphs with bounded densities of small subgraphs. J. Graph Theory, 29:185–207, 1998.

[Haj65] A. Hajnal. A theorem on k-saturated graphs. Can. J. Math., 17:720– 724, 1965.

[Han66] G. Hansel. Sur de nombre des fonctions Booléennes monotones de n variables. C. R. Acad. Sci. Paris Sér. A-B, 262:1088–1090, 1966.

[HS84] D. Hanson and K. Seyffarth. k-saturated graphs of prescribed maximum degree. Congres. Numer., 42:169–182, 1984.

[HT87] D. Hanson and B. Toft. Edge-colored saturated graphs. J. Graph Theory, 11:191–196, 1987.

[HT91] D. Hanson and B. Toft. k-saturated graphs of chromatic number at least k. Ars Combinatoria, 31:159–164, 1991.

[Isa75] R. Isaacs. Infinite families of non-trivial trivalent graphs which are not Tait colorable. Amer. Math. Monthly, 82:221–239, 1975.

[Kal84] G. Kalai. Weakly saturated graphs are rigid. Ann. Discrete Math., 20:189–190, 1984.

[Kal85] G. Kalai. Hyperconnectivity of graphs. Graphs Combin., 1:65–79, 1985.

[Kal90] G. Kalai. Symmetric matroids. J. Combin. Theory (B), 50:54–64, 1990.

[Kat66] G. O. H. Katona. A theorem on finite sets. In Theory of Graphs. Proc.

Colloq. Tihany, pages 187–207. Akademiai Kiado. Academic Press, New York, 1966.

[Kat74] G. O. H. Katona. Extremal problems for hypergraphs. In Combinatorics, volume 56, pages 13–42. Math. Cent. Tracts, 1974.

[KNS64] G. O. H. Katona, T. Nemetz, and M. Simonovits. On a graph problem of Turán (In Hungarian). Mat. Fiz. Lapok, 15:228–238, 1964.

[Kor81] A. D. Korshunov. The number of monotone Boolean functions. Probl.

Kibernet., 38:5–108, 1981.

[Kos82] A. V. Kostochka. A class of constructions for Turán (3, 4)-problem.

Combinatorica, 2:187–192, 1982.

[Kru63] J. B. Kruskal. The number of simplices in a complex. In R. Bellman, editor, Mathematical Optimization Techniques, pages 251–278. Univ. California Press, Berkeley, 1963.

[KS67] G. O. H. Katona and E. Szemerédi. On a problem of graph theory.

Studia Sci. Mat. Hung., 2:23–28, 1967.

[KS79] D. J. Kleitman and J. B. Shearer. Probabilities of independent choices being ordered. Stud. Appl. Math., 60:271–276, 1979.

[KT86] L. Kászonyi and Z. Tuza. Saturated graphs with minimal number of edges. J. Graph Theory, 10:203–210, 1986.

[KT96] K. M. Koh and C. P. Teo. Chromaticity of series-parallel graphs. Discrete Math., 154:289–295, 1996.

[Lip78] W. Lipski. On strings containing all subsets as substrings. Discrete Math., 21:253–259, 1978.

[Lov77] L. Lovász. Flats in matroids and geometric graphs. In P. J. Cameron, editor, Proc. 6th British Combin. Conf., pages 45–86. Academic Press, 1977.

[Mar75] M. Marcus. Finite Dimensional Multilinear Algebra, I, II. Marcel Dekker, New York, 1973, 1975.

[Moo69] J. W. Moon. The number of labeled k-trees. J. Combin. Theory, 6:196– 199, 1969.

[MRS91] M. Miller, I. Roberts, and J. Simpson. Application of symmetric chains to an optimization problem in the security of statistical databases. Bull. ICA, 2:47–58, 1991.

[MS95] K. C. Miller and D. G. Sarvate. An application of symmetric chains to a statistical database compromise prevention problem. Bull. ICA, 13:57–64, 1995.

[Oll72] L. T. Ollman. K_{2,2}-saturated graphs with a minimal number of edges.

In Combinatorics, Graph Theory and Computing, Proc. 13th Southeast. Conf., Boca Raton, (Utilitas Math., Winnipeg), pages 367–392, 1972.

[Oxl92] J. G. Oxley. Matroid Theory. Oxford Univ. Press, 1992.

[Pen93] Y.-H. Peng. On the chromatic uniqueness of certain trees of polygons.

J. Austral. Math. Soc. Ser. A, 55:403–410, 1993.

[Pik97] O. Pikhurko. On edge decomposition of posets. Submitted to Order, 1997.

[Pik98] O. Pikhurko. Uniform families and count matroids. Submitted to Graphs Comb., 1998.

[Pik99a] O. Pikhurko. Application of gross matroids to w-sat-problems. In preparation, 1999.

[Pik99b] O. Pikhurko. Asymptotic evaluation of the sat-function for r-stars. To appear in Discrete Math., 1999.

[Pik99c] O. Pikhurko. Enumeration of labelled (k, m)-trees. J. Combin. Theory (A), 86:197–199, 1999.

[Pik99d] O. Pikhurko. The minimal size of saturated hypergraphs. To appear in Combin. Prob. Comput., 1999.

[Pik99e] O. Pikhurko. On the triangle-vs-star size Ramsey number. Submitted to Combinatorica, 1999.

[PW70] M. J. Piff and D. J. A. Welsh. On the vector representation of matroids.

J. Lond. Math. Soc., 2:284–288, 1970.

[Röd85] V. Rödl. On a packing and covering problem. Europ. J. Combin., 5:69–78, 1985.

[RR70] C. Rényi and A. Rényi. The Prüfer code for k-trees. In P. Erdős et al., editors, Combinatorial Theory and Its Applications, pages 945– 971. North-Holland, Amsterdam, 1970.

[Sau73] N. Sauer. On the density of families of sets. J. Combin. Theory (A), 13:145–147, 1973.

[She72] S. Shelah. A combinatorial problem: Stability and order for models and theories in infinitary languages. Pac. J. Math, 4:247–261, 1972.

[Sid87] A. F. Sidorenko. Solution of a problem of Bollobás on 4-graphs. Mat.

Zametki, 41:433–455, 1987.

[Sid95] A. Sidorenko. What we know and what we do not know about Turán numbers. Graphs Combin., 11:179–199, 1995.

[ST98] R. H. Schelp and A. G. Thomason. A remark on the number of complete and empty subgraphs. Combin. Prob. Computing, 7:217–220, 1998.

[TT91] M. Truszczynski and Z. Tuza. Asymptotics results on saturated graphs.

Discrete Math., 87:309–314, 1991.

[Tuz86] Z. Tuza. A generalization of saturated graphs for finite languages. In Proc. IMYCS’86, Smolenice Castle (Czechoslovakia), volume 185 of MTA SZTAKI Studies, pages 287–293, 1986.

[Tuz88] Z. Tuza. Extremal problems on saturated graphs and hypergraphs. Ars Combinatoria, 25B:105–113, 1988.

[Tuz89] Z. Tuza. C_{4}-saturated graphs of minimum size. Acta Univ. Carolin.

Math. Phys., 30:161–167, 1989.

[Tuz92] Z. Tuza. Asymptotic growth of sparse saturated structures is locally determined. Discrete Math., 108:397–402, 1992.

[VC71] V. N. Vapnik and A. Chervonenkis. The uniform convergence of frequences of the appearance of events to their probabilities (in Russian). Teor. Veroyatn. Primen., 16:264–279, 1971.

[Ver99] J. Verstraëte. On arithmetic progressions of cycle lengths in graphs.

Preprint, 1999.

[Wel76] D. J. A. Welsh. Matroid Theory, volume 8 of London Math. Soc. Monographs. Academic Press, 1976.

[Wes66] W. Wessel. Über eine Klasse paarer Graphen, I: Beweis einer Vermutung von Erdős, Hajnal and Moon. Wiss. Z. Hochsch. Ilmenau, 12:253–256, 1966.

[Wes67] W. Wessel. Über eine Klasse paarer Graphen, II: Bestimmung der Minimalgraphen. Wiss. Z. Hochsch. Ilmenau, 13:423–426, 1967.

[Whi88] E. G. Whitehead, Jr. Chromatic polynomials of generalized trees. Discrete Math., 72:391–393, 1988.

[Whi89] W. Whiteley. A matroid on hypergraphs, with applications in scene analysis and geometry. Discrete Comput. Geom., 4:75–95, 1989.

[Whi96] W. Whiteley. Some matroids from discrete applied geometry. In J. E.

Bonin, J. G. Oxley, and B. Servatius, editors, Matroid Theory, volume 197 of Contemporary Mathematics, pages 171–311. AMS, Providence RI, 1996.

[WW84] N. White and W. Whiteley. A class of matroids defined on graphs and hypergraphs by counting properties. Preprint, Math. Dept., Univ. Florida, Gainesville, FL 32611, 1984.

[XWCY97] L. Xiaohui, J. Wenzhou, Z. Chengxue, and Y. Yuansheng. On smallest maximally nonhamiltonian graphs. Ars Combinatoria, 45:263–270, 1997.

[Yu93] J. Yu. An extremal problem for sets: A new approach via Bezoutians.

J. Combin. Theory (A), 62:170–175, 1993.

[Zyk49] A. A. Zykov. On some properties of linear complexes (in Russian).

Mat. Sbornik N.S., 24:163–188, 1949.

Index

()-representation, 96 (∗)-representation, 96 (k, m)-tree, vi, 113 H-built-tree, 117 α(n, t, k)-graph, 38 ex-function, 26 k-gon tree, vi, 114, 116 k-graph, i, 2 m-tree, 113 n-check, 125, 133 n × m-matrix, 18 n-set, 14 F-admissible, 2, 10, 19 F-closure, 24 F-free, 1, 2, 19 F-kills, 24 F-saturated, i, 1, 2, 10, 11, 19

acyclic hypergraph, 11

balanced set, 55 bisplit function, viii, 122

C-proof, 50, 63 c-proof, 50, 61 c-prove, 50, 61 centre, 30 chain, iv, 49, 60 chain decomposition, iv, 94 chord, 78 closure, 24 complementary chain, 104 cone, iv, 51, 85 consecutive partition, 64 count matroid, 58 cycle, 11 cycle-free hypergraph, 11

density, 3 design, 33 directed hypergraph, 11 dumb-bell, 27, 83

edge decomposition, 94, 95 edge partition, 95 exterior algebra, 52

G-proof, 50, 62 g-proof, 49, 61 g-prove, 49, 61 generalized star, ii, 30 generic position, 52 greedy algorithm, 124 gross matroid, 54

Hasse diagram, 95

hyperconnectivity matroid, 54

increasing function, 55 independent set, 7, 11, 15 integral function, 55

join, 90

kill, ii, 24 lap, 113 layer, 14 layered k-graph, 15 layered set, 14 left interior product, 53 line poset, v, 95 linear function, 56 local density, 3 local sparseness, 48 lower shadow, 8

m-proof, 61 m-prove, 61 m′-prove, 62 monotonically F-saturated, 2

ordered hypergraph, 14 orientation, 11 oriented n-cube, 96 orthogonal chains, 105 orthogonal partitions, 105 overlap size, 113

proper ordering, 47 proper subset, 55 pyramid, iii, 50, 64

r-proof, 61 r-prove, 61 rooted tree, 114, 117

saturation, i shift, 101 signature, 14, 15 simple matrix, 19 size Ramsey number, vii skipless chain, iv, 95 sparseness, 48 star, 5, 30 strongly F-saturated, 2 sub-design, 5, 33 submatrix, 19 submodular function, 55 support, 53 surplus, 28, 83 symmetric chain, v, 101 symmetric chain order, 101 symmetric matroid, 58, 90

tail, 8

tensor product, 90 transversal matroid, 58 triangle-free hypergraph, 39 triangular family, ii, 39 Turán function, 26 Turán number, 35

uniform family, iv, 5, 49, 69 uniform matroid, 63 upper shadow, 8, 15

vertex decomposition, 94, 95 vertex partition, 95

weak closure, 47 weak saturation, iii weakly F-saturated, iii, 1, 47 width, 95