Amenability and Ramsey theory in the metric setting

TL;DR

This paper extends Moore's characterization of amenability via Ramsey properties from ultrahomogeneous structures to metric Fraïssé structures, including all Polish groups, and shows that amenability is a G_delta property.

Contribution

It generalizes the connection between amenability and Ramsey properties to the metric setting, covering all Polish groups.

Findings

Amenability characterized by a Ramsey-type property in metric structures

Amenability is a G_delta condition in the topological space of groups

Extension of Moore's results to a broader class of groups

Abstract

Moore characterized the amenability of automorphism groups of countable ultrahomogeneous structures by a Ramsey-type property. We extend this result to automorphism groups of metric Fra\"iss\'e structures, which encompass all Polish groups. As an application, we prove that amenability is a G_delta condition.