Monadic second-order properties of very sparse random graphs

We study asymptotical probabilities of first order and monadic second order properties of Bernoulli random graph G(n,n−a). The random graph obeys FO (MSO) zero-one k-law (k is a positive integer) if, for any first order (monadic second order) formulae with quantifier depth at most k, it is true for G(n,n−a) with probability tending to 0 or to 1. Zero-one k-laws are well studied only for the first order language and a<1. We obtain new zero-one k-laws (both for first order and monadic second order languages) when a>1. Proofs of these results are based on the earlier studies of first order equivalence classes and our study of monadic second order equivalence classes. The respective results are of interest by themselves.