On limit points of spectra of first-order sentences with quantifier depth 4

Abstract

We study the asymptotic behavior of probabilities of first-order properties of sparse binomial random graphs. We consider properties with quantifier depth not more than 4. It is known that the only possible limit points of the spectrum (i.e., the set of all positive $\alpha$ such that $G\left(n,n^{-\alpha }\right)$ does not obey the zero-one law with respect to the property) of such a property are 1/2 and 3/5. We prove that 1/2 is not a limit point of the spectrum.