Ramsey theory and topological dynamics for first order theories

TL;DR

This paper explores the connections between Ramsey theory, topological dynamics, and model theory, introducing properties for first order theories and characterizing their dynamical features, with implications for the structure of Ellis groups and Galois groups.

Contribution

It introduces new Ramsey-like properties for first order theories and characterizes them via dynamical properties, providing criteria for finiteness and triviality of Ellis groups, with concrete examples and computations.

Findings

Profinite Ellis groups imply profinite Kim-Pillay Galois groups.

Criteria for finiteness and triviality of Ellis groups are established.

Counterexamples show lack of implications between fundamental properties.

Abstract

We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. As an initial motivation, we note that profiniteness of the Ellis group of a theory implies that the Kim-Pillay Galois group of this theory is also profinite, which in turn is equivalent to the equality of the Shelah and Kim-Pillay strong types. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelski's conjecture for amenable theories (proved in [15]) cannot be dropped.