Abstract
In 1998, Cherlin classified all homogeneous countable directed graphs (which are exactly the ones that appear as Fraisse limits). In the style of the Kechris-Pestov-Todorcevic correspondence of 2005 we examine amenability and unique ergodicity of the automorphism groups of these directed graphs by examining the combinatorics of their finite subgraphs.

In joint work with Miodrag Sokic (York), by adapting techniques of Angel-Kechris-Lyons and Zucker, we were able to establish amenability and unique ergodicity for the automorphism groups of all but one graph from Cherlin's classification.

This talk will focus on these results in the setting of tournaments, n-partite directed graphs and the so-called "semi-generic" digraph (which is the troublemaker of the bunch).