Large simplicial complexes: Universality, Randomness, and Ampleness

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.