Amenability and Ramsey theory in the metric setting

In recent years, there has been a flurry of activity relating notions linked to amenability or groups on one side, and combinatorial conditions linked to Ramsey theory on the other side. In this paper, we extend a result of Moore ([M2, theorem 7.1]) on the amenability of closed subgroups of S∞ to general Polish groups. A topological group is said to be amenable if every continuous action of the group on a compact space admits an invariant probability measure. Moore’s result is the counterpart of a theorem of Kechris, Pestov and Todorčević ([KPT]) on extreme amenability. A topological group is said to be extremely amenable if every continuous action of the group on a compact space admits a fixed point. In the context of closed subgroups of S∞, which are exactly the automorphism groups of Fraïssé structures, Kechris, Pestov and Todorčević characterize extreme amenability by a combinatorial property of the associated Fraïssé class (in the case where its objects are rigid), namely, the Ramsey property. A class K of structures is said to have the Ramsey property if for all structures A and B in K, for all integers k, there is a structure C in K such that for every coloring of the set of copies of A in C with k colors, there exists a copy of B in C within which all copies of A have the same color. Thus, extreme amenability, which provides fixed points, corresponds to colorings having a "fixed", meaning monochromatic, set. Amenability, on the other side, provides invariant measures. Since a measure is not far from being a barycenter of point masses, the natural mirror image of the Ramsey property in that setting should be for a coloring to have a "monochromatic convex combination of sets". Indeed, Tsankov (in an unpublished note) and Moore introduced a convex Ramsey property and proved that a Fraïssé class has the convex Ramsey property if and only if the automorphism group of its Fraïssé limit is amenable. Besides, Kechris, Pestov and Todorčević’s result was extended to general Polish groups by Melleray and Tsankov in [MT1]. They use the framework of continuous logic (see [BYBHU]) via the observation that every Polish group is the automorphism group of an approximately homogeneous metric structure ([M1]), that is of a metric Fraïssé limit (in the sense of [BY]). They define an approximate Ramsey property for classes of metric structures and then show that a metric Fraïssé class has the approximate Ramsey property if and only if the automorphism group of its Fraïssé limit is extremely amenable. In this paper, we "close the diagram" by giving a metric version of Moore’s result. We replace the classical notion of a coloring with the metric one (from [MT1]) to define a metric convex Ramsey property, and we prove the exact analogue of Moore’s theorem: