Sparse graphs and the fixed points on type spaces property

TL;DR

This paper investigates the fixed points on type spaces property in omega-categorical sparse graphs derived from Hrushovski constructions, revealing limitations in expansions maintaining this property.

Contribution

It demonstrates that certain omega-categorical structures from Hrushovski constructions lack expansions with the fixed points on type spaces property.

Findings

Existence of omega-categorical structures without fixed points property in any expansion

Extension of previous results by Evans, Hubicka, and Nesetril

Insights into topological dynamics of automorphism groups of sparse graphs

Abstract

We examine the topological dynamics of the automorphism groups of omega-categorical sparse graphs resulting from Hrushovski constructions. Specifically, we consider the fixed points on type spaces property, which a structure M has if, for each positive integer n, every Aut(M)-subflow of the space of n-types has a fixed point. Extending a result of Evans, Hubicka and Nesetril, we show that there exists an omega-categorical structure M, resulting from a Hrushovski construction, such that no omega-categorical expansion of M has the fixed points on type spaces property.