Automorphism groups and Ramsey properties of sparse graphs

TL;DR

This paper explores the automorphism groups of sparse graphs, linking topological dynamics and Ramsey theory, and resolves an open question about the existence of certain highly symmetric expansions.

Contribution

It demonstrates that Hrushovski's sparse graph example lacks an $\, ext{omega}$-categorical expansion with an extremely amenable automorphism group, advancing understanding of automorphism groups in sparse graphs.

Findings

Identifies properties of automorphism groups of sparse graphs

Shows non-existence of certain $\, ext{omega}$-categorical expansions

Connects topological dynamics with structural Ramsey theory

Abstract

We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todor\v{c}evi\'c correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an $\omega$-categorical sparse graph has no $\omega$-categorical expansion with extremely amenable automorphism group.