The Rado simplicial complex

Michael Farber, Lewis Mead, Lewin Strauss

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)339-356
Number of pages18
JournalJournal of Applied and Computational Topology
Volume5
Issue number2
DOIs
StatePublished - Jun 2021
Externally publishedYes

Keywords

  • Ample simplicial complex
  • Rado graph
  • Rado simplicial complex
  • Random simplicial complex
  • Universal simplicial complex

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