Ramsey theory and topological dynamics for first order theories

Krupiński, Krzysztof; Lee, Junguk; Moconja, Slavko

doi:10.1090/tran/8594

Ramsey theory and topological dynamics for first order theories
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by Krzysztof Krupiński, Junguk Lee and Slavko Moconja PDF

Trans. Amer. Math. Soc. 375 (2022), 2553-2596 Request permission

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. 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 by Krupiński, Newelski, and Simon [J. Math. Log. 19 (2019), no. 2, 1950012, p. 55]) cannot be dropped.

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