Topological matchings and amenability
We establish a characterization of amenability for general Hausdorff topological groups in terms of matchings with respect to finite uniform coverings. Furthermore, we prove that it suffices to just consider two-element uniform coverings. We also show that extremely amenable as well as compactly approximable topological groups satisfy a perfect matching property condition—the latter even with regard to arbitrary (i.e., possibly infinite) uniform coverings. Finally, we prove that the automorphism group of a Fraïssé limit of a metric Fraïssé class is amenable if and only if the class has a certain Ramsey-type matching property.