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