Binary simple homogeneous structures are supersimple with finite rank

Suppose that M is an infinite structure with finite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous then its complete theory is supersimple with finite SU-rank which cannot exceed the number of complete 2-types over the empty set

On properties of (weakly) small groups

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary : a weakly small group with simple theory has an infinite definable finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal solvable group A of derived length n is contained in an A-definable almost solvable group of class n

Fields and rings with few types

Let R be an associative ring with possible extra structure. R is said to be weakly small if there are countably many 1-types over any finite subset of R. It is locally P if the algebraic closure of any finite subset of R has property P. It is shown here that a field extension of finite degree of a weakly small field either is a finite field or has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. Every weakly small division ring of positive characteristic is locally finite dimensional over its centre. The Jacobson radical of a weakly small ring is locally nilpotent. Every weakly small division ring is locally, modulo its Jacobson radical, isomorphic to a product of finitely many matrix rings over division rings

Expansions géométriques et ampleur

The main result of this thesis is the study of how ampleness grows in geometric and SU-rank omega structures when adding a new independent dense/codense subset. In another direction, we explore relations of ampleness with equational theories; there, we give a direct proof of the equationality of certain CM-trivial theories. Finally, we study indiscernible closed sets—which are closely related with equations—and measure their complexity in the free pseudoplaneLe résultat principal de cette thèse est l'étude de l'ampleur dans des expansions des structures géométriques et de SU-rang oméga par un prédicat dense/codense indépendant. De plus, nous étudions le rapport entre l'ampleur et l'équationalite, donnant une preuve directe de l'équationalite de certaines théories CM-triviales. Enfin, nous considérons la topologie indiscernable et son lien avec l'équationalite et calculons la complexité indiscernable du pseudoplan libr

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

On sets with rank one in simple homogeneous structures

We study definable sets $D$ of SU-rank 1 in $M^{eq}$, where $M$ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such $D$ can be seen as a `canonically embedded structure', which inherits all relations on $D$ which are definable in $M^{eq}$, and has no other definable relations. Our results imply that if no relation symbol of the language of $M$ has arity higher than 2, then there is a close relationship between triviality of dependence and $D$ being a reduct of a binary random structure. Somewhat more preciely: (a) if for every $n \geq 2$, every $n$-type $p(x_1, ..., x_n)$ which is realized in $D$ is determined by its sub-2-types $q(x_i, x_j) \subseteq p$, then the algebraic closure restricted to $D$ is trivial; (b) if $M$ has trivial dependence, then $D$ is a reduct of a binary random structure