TCC Course "Descriptive Combinatorics"

Short Description

The course will concentrate on the emerging field of descriptive combinatorics that studies definable graphs on topological spaces and their combinatorial-type invariants (for example, the minimum number of colours needed for a proper vertex-colouring provided each colour class is Borel, or Lebesgue measurable, or has the property of Baire). This field connects combinatorics on one hand with descriptive set theory, limits of dicrete structures, measure theory, group actions, etc, on the other hand, where applications go in both directions. The survey "Descriptive Graph Combinatorics" by Alexander S. Kechris and Andrew S. Marks gives a good overview of this area. This course will concentrate on those aspects where the interplay between combinatorics and other areas of mathematics is most prominent.

Tentative List of Main Topics

* = if time permits


Knowledge of the basics of graph theory, measure theory and topology will be assumed. No familiarity with descriptive set theory is required. Some suggested general references (e.g. if you need to look up a specific topic) are:


The lectures will take place 4-6pm on Thursdays, 17 January - 7 March 2019.

Content of lectures


Those participants who need a grade will have to write an essay presenting one of the suggested topics (with their list to be added here later).

The aim of each essay is to understand some further results and explain them in a way accessible to other course participants.

Getting Help

If you have any course-related questions, please post them at the discussion group of the course:!forum/desriptive-combinatorics

Also, other messages related to the course are welcome on the group. In particular, if you see a question by somebody else and you know the answer or just want to add a comment, you are very welcome to post a follow-up message.

In order to join the group (which is open to members only), you can either request this via the group's webpage or email me.

Forthcoming Related Meetings in the UK

This course should be a useful preparation if you intend to take part in any of these meetings (especially for the second one):
  • Number Theory and Dynamics Conference, Cambridge, 25-29 March 2019
  • Workshop "Measurability, Ergodic Theory and Combinatorics", Warwick, 8-12 July 2019

  • Oleg Pikhurko (O dot Pikhurko at warwick dot ac dot uk)