Omega-categorical simple theories

This thesis touches on many different aspects of homogeneous relational structures. We start with an introductory chapter in which we present all the background from model theory and homogeneity necessary to understand the results in the main chapters. The second chapter is a list of examples. We present examples of binary and ternary homogeneous relational stuctures, and prove the simplicity or non-simplicity of their theory. Many of these examples are well-known structures (the ordered rational numbers, random graphs and hypergraphs, the homogeneous Kn-free graphs), while others were constructed during the first stages of research. In the same chapter, we present some combinatorial results, including a proof of the TP2 in the Fraïssé limit of semifree amalgamation classes in the language of n-graphs, such that all the minimal forbidden configurations of the class of size at least 3 are all triangles. The third chapter contains the main results of this thesis. We prove that supersimple finitely homogeneous binary relational structures cannot have infinite monomial SU-rank, show that primitive binary supersimple homogeneous structures of rank 1 are “random” in the sense that all their minimal forbidden configurations are of size at most 2, and partially classify the supersimple 3-graphs under the assumption of stable forking in the theories of finitely homogeneous structures with supersimple theory. The fourth chapter is a proof of the directed-graph version of a well-known result by Erdős, Kleitman and Rothschild. Erdős et al. prove that almost all finite labelled trianglefree simple graphs are bipartite, and we prove that almost all finite labelled directed graphs in which any three distinct vertices span at least one directed arc consist of two disjoint tournaments, possibly with some directed arcs from one to the other.