  In this paper we state the homological domination principle for random
multi-parameter simplicial complexes, claiming that the Betti number in one
specific dimension (which is explicitly determined by the probability
multi-parameter) significantly dominates the Betti numbers in all other
dimensions. We also state and discuss evidence for two interesting conjectures
which would imply a stronger version of the homological domination principle,
namely that generically homology of a random simplicial complex coincides with
that of a wedges of k-dimensional spheres. These two conjectures imply that
under an additional assumption (specified in the paper) a random simplicial
complex collapses to a k-dimensional complex homotopy equivalent to a wedge of
spheres of dimension k.
