Some interactions between model theory and combinatorics

In the research area of model theory, one is concerned with what can be said about certain mathematical structures using first-order sentences and what sets can be defined within these structures using first-order formulas. A big part of the project involves the identification of various classes of well-behaved structures. One hopes to prove interesting theorems that apply to every member of the class.

Often, the definition of such a class has a combinatorial flavour. Furthermore, there are several instances of results from combinatorics proving useful in the study of these classes.

There are applications in the other direction too, for example where techniques based in model theory are used to prove results about certain families of graphs.

In this talk, I shall give a brief survey of some of the points of contact between model theory and combinatorics and then focus on one particular combinatorial result, known as the (p,q)-theorem, that has helped with the development of model theoretic ideas and continues to inspire further work. That part of the talk will be based on joint work with Tsinjo Rakotonarivo and Charlotte Kestner.