Ample simplicial 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 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 bigl (Omega bigl (frac{2^r}{sqrt{r}}bigr )bigr ). 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.