Ample simplicial complexes

Abstract

AbstractMotivated 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 $$\exp \bigl (\Omega \bigl (\frac{2^r}{\sqrt{r}}\bigr )\bigr )$$ exp ( Ω ( 2 r r ) ) . We use the probabilistic method to establish the existence of r-ample simplicial complexes with n vertexes for any $$n>r 2^r 2^{2^r}$$ 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.