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.
Highlights
The article [KPT05] by Kechris, Pestov and Todorcevic established a fundamental correspondence between structural Ramsey theory and topological dynamics
[KPT05] provided an extremely powerful tool to compute an invariant known as the universal minimal flow, and immediately led to an explicit representation of this invariant in many concrete cases, including the case of most homogeneous graphs
Having constructed Ramsey precompact expansions of all homogeneous directed graphs and having verified the expansion property in each case, we are automatically given a list of the respective universal minimal flows
Summary
The article [KPT05] by Kechris, Pestov and Todorcevic established a fundamental correspondence between structural Ramsey theory and topological dynamics. The ordering property technique was generalized in [NVT13] to the notion of precompact expansion, opening the door for further applications. This was in particular used to compute the universal minimal flows of the automorphism groups of the homogeneous circular directed graphs S(2) and S(3). The purpose of this paper is to apply these new methods in the context of homogeneous directed graphs; following the classification by Cherlin in [C98], we show that in each case its age allows a Ramsey precompact expansion. It will be shown that in addition, this expansion can be used to compute the universal minimal flow of the corresponding automorphism group. C −→ (B)Ak,l to mean that for every map c :
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