Erdös Lap Number
Definition:
 Paul Erdös has Erdös Lap Number 0.
 In order to acquire Erdös Lap Number i one has to sit on
a lap of a person with the number i1.
NB: A proof (in a form of a
photo or a notary certification) is required for any claims.
For example, Barbie Freidin
(née Benzer) has the Erdös Lap Number of 1. On this photo
(taken from the Fall
1996
issue
of
"Math PUrview") she is the leftmost epsilon. 

Therefore, Oleg Pikhurko gets
the the Erdös Lap Number 2. On the photo he is the O(1) (the O big
of one) sitting on Barbie's lap.


Open Question (Małgorzata Bednarska
$10): Does there exist a person whose
(usual) Erdös
Number = Erdös Lap Number =
1 ?
Peter Griztmann conjectured that the answer is "Yes" and that
János Pach is such a person. Unfortunately, even János
himself was not able to produce a proof that his Erdös Lap Number
is 1 (although he said that "it is a
reasonable conjecture").
The following photo (posted
by
János
Pach on Gil Kalai's blog) led some people to
believe that they have a promising approach to the proof.
However, Uncle Paul holds not János but his daughter...
The best known result towards the conjecture is that there is a person with Erdös Number 1 and Erdös Lap Number 2 (namely, Norbert Hegyvári):
Theorem (Ron Graham): The Erdös lap
digraph contains a directed cycle!
Proof
(presented by Ron at the Erdös Centennial Conference, 2013).
Further developments coming that direction:
The Erdös Lap Graph is
expanding!...
Erdos lap number 3 comes as an extra bonus for 21484 "Graph Theory"
students:
Sometimes I'm recognized even in public transport :)