Large simplicial complexes: Universality, Randomness, and Ampleness

TL;DR

This paper surveys recent advances in the study of large simplicial complexes, focusing on properties like ampleness, universality, and randomness, and introduces new results on their topological complexity and existence.

Contribution

It introduces the concept of $r$-ample simplicial complexes, discusses their constructions, and presents new results on the topological complexity of random complexes and the unique $ ext{infinite}$-ample (Rado) complex.

Findings

Random simplicial complexes in the medial regime are ample.

The topological complexity of these complexes is at most 4 with high probability.

There exists a unique $ ext{infinite}$-ample (Rado) complex on countable vertices.

Abstract

The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality. In the first part of the paper we concentrate on $r$-ample simplicial complexes which are high dimensional analogues of the $r$-e.c. graphs introduced originally by Erd\H os and R\'eniy. The class of $r$-ample complexes is useful for applications since these complexes allow extensions of subcomplexes of certain type in all possible ways; besides, $r$-ample complexes exhibit remarkable robustness properties. We discuss results about the existence of $r$-ample complexes and describe their probabilistic and deterministic constructions. The properties of random simplicial complexes in medial regime are important for this discussion since these complexes are ample, in certain range. We prove that the topological complexity of a random simplicial complex in the medial regime satisfies ${\sf TC}(X)\le 4$, with probability tending to $1$ as $n\to\infty$. There exists a unique (up to isomorphism) $\infty$-ample complex on countable set of vertexes (the Rado complex), and the second part of the paper surveys the results about universality, homogeneity, indestructibility and other important properties of this complex. The Appendix written by J.A. Barmak discusses connectivity of conic and ample complexes.