Title: Tournaments With Forbidden Substructures and the Erdos-Hajnal Conjecture
Description: A celebrated Conjecture of Erdos and Hajnal states that for every undirected graph H there exists ɛ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n^{ɛ(H)}. In 2001 Alon, Pach and Solymosi proved that the conjecture has an equivalent directed version, where undirected graphs are replaced by tournaments and cliques and stable sets by transitive subtournaments. This dissertation addresses the directed version of the conjecture and some problems in the directed setting that are closely related to it. For a long time the conjecture was known to be true only for very specific small graphs and graphs obtained from them by the so-called substitution procedure proposed by Alon, Pach and Solymosi. All the graphs that are an outcome of this procedure have nontrivial homogeneous sets. Tournaments without nontrivial homogeneous sets are called prime. They play a central role here since if the conjecture is not true then the smallest counterexample is prime.

We remark that for a long time the conjecture was known to be true only for some prime graphs of order at most 5. There exist 5-vertex graphs for which the conjecture is still open, however one of the corollaries of the results presented in the thesis states that all tournaments on at most 5 vertices satisfy the conjecture. In the first part of the thesis we will establish the conjecture for new infinite classes of tournaments containing infinitely many prime tournaments. We will first prove the conjecture for so-called constellations. It turns out that almost all tournaments on at most 5 vertices are either constellations or are obtained from constellations by substitutions. The only 5-vertex tournament for which this is not the case is a tournament in which every vertex has outdegree 2. We call this the tournament C_{5}. Another result of this thesis is the proof of the conjecture for this tournament. We also present here the structural characterization of the tournaments satisfying the conjecture in almost linear sense.

In the second part of the thesis we focus on the upper bounds on coefficients epsilon(H) for several classes of tournaments. In particular we analyze how they depend on the structure of the tournament. We prove that for almost all h-vertex tournaments ɛ(H) ≤ 4/h(1+o(1)). As a byproduct of the methods we use here, we get upper bounds for ɛ(H) of undirected graphs. We also present upper bounds on ɛ(H) of tournaments with small nontrivial homogeneous sets, in particular prime tournaments. Finally we analyze tournaments with big ɛ(H) and explore some of their structural properties.