Spectra of first-order formulas with a low quantifier depth and a small number of quantifier alternations

Resumo

Spectra of first-order formulas are studied. The spectrum of a first-order formula is the set of all positive α such that either this formula is true for the random graph G(n, n^−α) with an asymptotic probability being neither 0 nor 1 or the limit does not exist. It is well known that there exists a first-order formula with an infinite spectrum. The minimum number of quantifier alternations in such a formula is found.