Universal minimal flows of automorphism groups

A. S. Kechris, V. Pestov, S. Todorčević

We investigate some connections between the Fra\'{\i}ssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the $\infty$-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra,etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory, the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.