Ample simplicial complexes

TL;DR

This paper investigates $r$-ample simplicial complexes, demonstrating their topological properties, existence bounds, and explicit constructions, motivated by applications in network theory and computer science.

Contribution

It introduces the concept of $r$-ample simplicial complexes, proves their connectivity properties, establishes bounds on their size, and constructs explicit examples with near-optimal vertex counts.

Findings

$r$-ample complexes are simply connected and 2-connected for large $r$

Existence of $r$-ample complexes with exponentially many vertices

Explicit construction of nearly optimal $r$-ample complexes

Abstract

Motivated by potential applications in network theory, engineering and computer science, we study $r$-ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of {\it indestructibility,} in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an $r$-ample simplicial complex is simply connected and $2$-connected for $r$ large. The number $n$ of vertexes of an $r$-ample simplicial complex satisfies $\exp(\Omega(\frac{2^r}{\sqrt{r}}))$. We use the probabilistic method to establish the existence of $r$-ample simplicial complexes with $n$ vertexes for any $n>r 2^r 2^{2^r}$. Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed $r$-ample simplicial complexes with nearly optimal number of vertexes.