  We consider the multi-parameter random simplicial complex as a higher
dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate
appearance of "unusual" topological structures in the complex from the point of
view of large deviations. We first study upper tail large deviation
probabilities for subcomplex counts, deriving the order of magnitude of such
probabilities at the logarithmic scale precision. The obtained results are then
applied to analyze large deviations for the number of simplices at the critical
dimension and below. Finally, these results are also used to deduce large
deviation estimates for Betti numbers of the complex in the critical dimension.
