Ramsey Precompact Expansions of Homogeneous Directed Graphs

Abstract

In 2005, Kechris, Pestov and Todorčević provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow, immediately leading to an explicit representation of this invariant in many concrete cases. More recently, the framework was generalized allowing for further applications, and the purpose of this paper is to apply these new methods in the context of homogeneous directed graphs.In this paper, we show that the age of any homogeneous directed graph allows a Ramsey precompact expansion. Moreover, we verify the relative expansion properties and consequently describe the respective universal minimal flows.

Ordered structures and large conjugacy classes

“…Directed ultrahomogeneous graphs were classified by Cherlin [2] and their precompact Ramsey expansions were described by Jasiński-Laflamme-Nguyen van Thé-Woodrow [7]. See page 74 in [2] for the list of them, see also page 5 in [7].…”

“…By Kechris-Pestov-Todorcevic [11] and Nguyen Van Thé [13], automorphism groups of those expansions are extremely amenable. The notation we will use comes from [7]. Sketch of the proof.…”

“…3. The precompact Ramsey expansion P(3) * of the 'twisted' universal ordered poset P(3) is first-order simply bi-definable to the Fraïssé limit of the family K 0 of ordered posets, additionally equipped with 3 subsets (described using unary predicates) forming a partition of the universe of the ordered poset, see the bottom of the page 21 in [7]. Proposition 4.1 also applies to the age of S * , the precompact Ramsey expansion of the semigeneric directed graph S, which is rather straightforward to check.…”

“…Corollary 5. 7. Suppose that F is a full order expansion such that F − has free amalgamation, or the class of finite ordered tournaments.…”

Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs

“…We will work systematically through the classification of countable homogeneous directed graphs as given by [6]. In [10] this classification was used to work out the universal minimal flows explicitly. In a number of cases (some already exploited by [26]) the appropriate Ramsey class is not obtained from expansions by orders, so we will need to reformulate the methods of [3] in a somewhat broader setting.…”