Amenability and Ramsey Theory

TL;DR

This paper links the concept of amenability in discrete groups with a new form of structural Ramsey theory, offering a weaker criterion and revealing properties of non-amenable groups related to finitely additive measures.

Contribution

It introduces a Ramsey-theoretic reformulation of amenability, providing a novel perspective that weakens the Folner criterion and explores measure properties in non-amenable groups.

Findings

Ramsey-theoretic reformulation of amenability

Existence of subsets in non-amenable groups with specific measure properties

Weakening of the Folner criterion for amenability

Abstract

The purpose of this article is to connect the notion of the amenability of a discrete group with a new form of structural Ramsey theory. The Ramsey theoretic reformulation of amenability constitutes a considerable weakening of the Folner criterion. As a by-product, it will be shown that in any non amenable group G, there is a subset E of G such that no finitely additive probability measure on G measures all translates of E equally.