First order sentences about random graphs: Small number of alternations

The spectrum of a first order sentence is the set of all α such that G(n,n−α) does not obey zero-one law with respect to this sentence. In this paper, we prove that the minimal number of quantifier alternations of a first order sentence with infinite spectrum equals 3. We have also proved that the spectrum of a first-order sentence with quantifier depth 4 has no limit points except possibly the points 1∕2 and 3∕5.