The Ramsey property for Banach spaces and Choquet simplices

We show that the Gurarij space $\mathbb{G}$ has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. also compute universal minimal flow group Poulsen simplex $\mathbb{P}$ we prove it consists canonical action on itself. Conley Törnquist. pointwise stabilizer any closed proper face is amenable. Similarly, biface unit ball dual (the Lusky simplex) These results are obtained via several Kechris–Pestov–Todorcevic correspondences, by establishing approximate Ramsey property for classes finite-dimensional Banach spaces function systems their versions with distinguished contractions. first direct application correspondence in setting metric structures. The fundamental combinatorial principle underpins proofs Dual Theorem Graham Rothschild.