Zero-one k-law

In this article we study asymptotical behavior of the probabilities of some properties of Erd?s–Rényi random graphs $G(N,p)$. We consider the first-order properties and the probabilities $p=N^{?\alpha }$ for rational $\alpha$. The zero-one law in ordinary sense for these graphs doesn’t hold. We weakened the law by considering the formulas with quantifier depth bounded by a fixed number. To prove our results we used theorems on estimates for the number of extensions of small subgraphs in the random graph. Such an approach was first used by Spencer and Shelah in 1988. We also used our recent results from this area.