The interplay between structural Ramsey theory and topological dynamics of automorphism groups has been extensively studied since their connection was established in a paper by Kechris-Pestov-Todorcevic, while earlier works of Pestov, and Glasned and Weiss exhibited the phenomena in special cases. This line of research was extended to metric structures and approximate Ramsey property by Melleray and Tsankov. We establish the approximate Ramsey property for the class of finite-dimensional normed vector spaces and deduce that the group of linear isometries of the universal approximately homogeneous Banach space, the Gurarij space, is extremely amenable, that is, every continuous action on a compact Hausdorff space has a fixed point. Dualizing our ideas, we show that the class of finite-dimensional simplexes with a distinguished extreme point and affine surjections satisfies the approximate Ramsey property. As a consequence, we find that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is its natural action on the Poulsen simplex. This is a joint work (in progress) with Aleksandra Kwiatkowska (UCLA), Jordi Lopez Abad (ICMAT Madrid and USP) and Brice Mbombo (USP).
