The Rado Simplicial Complex

TL;DR

The paper introduces the Rado simplicial complex, a highly symmetric, universal, and unique countable complex that contains all countable complexes as subcomplexes, with applications in topology and combinatorics.

Contribution

It defines the Rado simplicial complex, proves its uniqueness, provides explicit constructions, and shows its properties and prevalence in random complexes.

Findings

The Rado complex is unique up to isomorphism.

A random simplicial complex is almost surely a Rado complex.

The geometric realisation of the Rado complex is contractible and homeomorphic to an infinite-dimensional simplex.

Abstract

A Rado simplicial complex X is a generalisation of the well-known Rado graph. X is a countable simplicial complex which contains any countable simplicial complex as its induced subcomplex. The Rado simplicial complex is highly symmetric, it is homogeneous: any isomorphism between finite induced subcomplexes can be extended to an isomorphism of the whole complex. We show that the Rado complex X is unique up to isomorphism and suggest several explicit constructions. We also show that a random simplicial complex on countably many vertices is a Rado complex with probability 1. The geometric realisation |X| of a Rado complex is contractible and is homeomorphic to an infinite dimensional simplex. We also prove several other interesting properties of the Rado complex X, for example we show that removing any finite set of simplexes of X gives a complex isomorphic to X.