Threshold spectra for random graphs

Abstract

Let G = G(n, p) be the random graph with n vertices and edge probability p and ƒ(n, p, A) be the probability that G has A, where A is a first order property of graphs. The evolution of the random graph is discussed in terms of a spectrum of p = p(n) where ƒ(n, p, A) changes. A partial characterization of possible spectra is given. When p = n-a, a irrational, and A is any first order statement, it is shown that lim ƒ(n, p, A) = 0 or 1.