Abstract
We study a remarkable simplicial complex X on countably many vertexes. X is universal in the sense that any countable simplicial complex is an induced subcomplex of X. Additionally, X is homogeneous, i.e. any two isomorphic finite induced subcomplexes are related by an automorphism of X. We prove that X is the unique simplicial complex which is both universal and homogeneous. The 1-skeleton of X is the well-known Rado graph. We show that a random simplicial complex on countably many vertexes is isomorphic to X with probability 1. We prove that the geometric realisation of X is homeomorphic to an infinite dimensional simplex. We observe several curious properties of X, for example we show that X is robust, i.e. removing any finite set of simplexes leaves a simplicial complex isomorphic to X. The robustness of X leads to the hope that suitable finite approximations of X can serve as models for very resilient networks in real life applications. In a forthcoming paper (Even-Zohar et al. Ample simplicial complexes, arXiv:2012.01483, 2020) we study finite approximations to the Rado complex, they can potentially be useful in real life applications due to their structural stability.
Highlights
Urysohn constructed a remarkable complete, separable metric space U, which is known as the Urysohn space
The Urysohn space U is homogeneous in the sense that any partial isometry between its finite subsets can be extended to a global isometry
In this paper we study a high-dimensional generalisation of the Rado graph which we call the Rado simplicial complex X
Summary
Finite random simplicial complexes in the medial regime are subcomplexes of the Rado complex X induced on a random subset of n vertexes It was proven in Farber and Mead (2020) that, with probability tending to 1, such simplicial complexes are quite special; for example, they have dimension ∼ log ln n +log log ln n and have vanishing Betti numbers in dimensions ≤ log ln n + c, where c is a constant. Paper Brooke-Taylor and Testa (2013) applies the methods of mathematical logic and model theory to study the geometry of simplicial complexes; it uses language very different from ours. The Fraïssé limit construction, when applied to the class of all finite simplicial complexes, produces a simplicial complex F on countably many vertexes which is universal and homogeneous, i.e. it is a Rado complex in the terminology of this paper. The authors thank the anonymous referees for their helpful comments
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