TY - JOUR

T1 - The Rado simplicial complex

AU - Farber, Michael

AU - Mead, Lewis

AU - Strauss, Lewin

N1 - Publisher Copyright:
© 2021, The Author(s).

PY - 2021/6

Y1 - 2021/6

N2 - 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.

AB - 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.

KW - Ample simplicial complex

KW - Rado graph

KW - Rado simplicial complex

KW - Random simplicial complex

KW - Universal simplicial complex

UR - http://www.scopus.com/inward/record.url?scp=85125013354&partnerID=8YFLogxK

U2 - 10.1007/s41468-021-00069-z

DO - 10.1007/s41468-021-00069-z

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AN - SCOPUS:85125013354

VL - 5

SP - 339

EP - 356

JO - Journal of Applied and Computational Topology

JF - Journal of Applied and Computational Topology

SN - 2367-1726

IS - 2

ER -