A Ramsey Space of Infinite Polyhedra and the Random Polyhedron

In this paper we introduce a new topological Ramsey space P whose elements are infinite ordered polyhedra. The corresponding familiy AP of finite approximations can be viewed as a class of finite structures. It turns out that the closure of AP under isomorphisms is the class KP of finite ordered polyhedra. Following [8], we show that KP is a Ramsey class. Then, we prove a universal property for ultrahomegeneous polyhedra and introduce the (ordered) random polyhedron, and prove that it is the Fräıssé limit of KP; hence the group of automorphisms of the ordered random polyhedron is extremely amenable (this fact is deduced from results of [6]). Later, we present a countably infinite family of topological Ramsey subspaces of P; each one determines a class of finite ordered structures which turns out to be a Ramsey class. One of these subspaces is Ellentuck’s space; another one is associated to the class of finite ordered graphs whose Fräıssé limit is the random graph. The Fräıssé limits of these classes are not pairwise isomorphic as countable structures and none of them is isomorphic to the random polyhedron. Finally, following [6], we calculate the universal minimal flow of the (non ordered) random polyhedron as well as the universal minimal flows of the (non ordered) random structures associated to our family of topological Ramsey subspaces of P.