Topological dynamics of unordered Ramsey structures

Abstract

We investigate the connections between Ramsey properties of Fraisse classes K and the universal minimal flow M(GK) of the automorphism group GK of their Fraisse limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class K has finite Ramsey degree for embeddings, then this degree equals the size of M(GK). We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if K is a relational Ramsey class and GK is amenable, then M(GK) admits a unique invariant Borel probability measure that is concentrated on a unique generic orbit.