Amenability and Ramsey theory
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 Følner 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. The analysis of discrete groups will be generalized to the setting of automorphism groups of ultrahomogeneous structures.