On the limit values of probabilities for the first order properties of graphs

Abstract

Consider the random graph ${\cal G}(n,p),$ where $p=p(n)$ is any threshold function satisfying $p(n) = \Theta(\ln n / n).$ We give a full characterization of the limit values of probabilities of ${\cal G}(n,p)$ having a property $\psi,$ where $\psi$ is any sentence of the first order theory of graphs.