  In this paper we study the notion of critical dimension of random simplicial
complexes in the general multi-parameter model described in our previous papers
of this series. This model includes as special cases the
Linial-Meshulam-Wallach model as well as the clique complexes of random graphs.
We characterise the concept of critical dimension in terms of various geometric
and topological properties of random simplicial complexes such as their Betti
numbers, the fundamental group, the size of minimal cycles and the degrees of
simplexes. We mention in the text a few interesting open questions.
