A Ramsey space of infinite polyhedra and the random polyhedron

TL;DR

This paper introduces a new topological Ramsey space of infinite ordered polyhedra, establishes Ramsey properties for finite polyhedra, and explores the automorphism group of the random polyhedron, revealing multiple Ramsey classes.

Contribution

It constructs a novel topological Ramsey space for infinite polyhedra and analyzes the Ramsey properties and automorphism groups of related structures.

Findings

The set of finite polyhedra satisfies two types of Ramsey properties.

The automorphism group of the random polyhedron is extremely amenable.

Multiple topological Ramsey subspaces correspond to distinct classes of finite structures.

Abstract

In this paper we introduce a new topological Ramsey space whose elements are infinite ordered polyhedra. Then, we show as an application that the set of finite polyhedra satisfies two types of Ramsey property: one, when viewed as a category over $\mathbb N$; the other, when considered as a class of finite structures. The (ordered) random polyhedron is the Fraisse limit of the class of finite ordered polyhedra; we prove that its group of automorphisms is extremely amenable. Finally, we present a countably infinite family of topological Ramsey subspaces; 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 Fraisse limit is the random graph. The Fraisse limits of these classes are not pairwise isomorphic as countable structures and none of them is isomorphic to the random polyhedron.