Quantifier alternation in first-order formulas with infinite spectra

The spectrum of a first-order formula is the set of numbers $\alpha$ such that for a random graph in a binomial model where the edge probability is a power function of the number of graph vertices with exponent $-\alpha$ the truth probability of this formula does not tend to either zero or one. In 1990 J. Spenser proved that there exists a first-order formula with an infinite spectrum. We have proved that the minimum quantifier depth of a first-order formula with an infinite spectrum is either 4 or 5. In the present paper we find a wide class of first-order formulas of depth 4 with finite spectra and also prove that the minimum quantifier alternation number for a first-order formula with an infinite spectrum is 3.