
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.
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