Model-theoretic considerations in finite combinatorics

This thesis contains a collection of results that lie primarily in the intersection of model theory and finite combinatorics. The two main areas of combinatorics that make an appearance in the pages to follow are Structural Ramsey Theory and Extremal Graph Theory.

Chapter 3 focusses on interactions between model theory and structural Ramsey theory. There, I essentially present two new results. The first is an extension of Scow’s theorem connecting generalised indiscernibles and Ramsey classes, and the second is a construction of a strict hierarchy of “local” Ramsey classes.

In Chapter 4, the focus switches, momentarily, to pure model theory, and specifically Classification Theory, in the sense of Shelah. This chapter investigates dividing lines characterised by collapses of indiscernibles and transfer principles for products of structures for such dividing lines. Structural graph theory also shows up in the last section, dealing with bounds on the twin-width of classes
of products of graphs.

In Chapter 5, I present results on extremal graph theory, in particular, Zarankiewicz’s problem, restricted to model-theoretically tame contexts. These include semibounded o-minimal structures, models of Presburger arithmetic, and some Hrushovski constructions.

Finally, there is a chapter with two more minor results that both fall, at least in my opinion, within the broader context of intersections of model theory and combinatorics, which closes the thesis.