  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.
