The first order convergence law fails for random perfect graphs

Abstract

We consider first order expressible properties of random perfect graphs. That is, we pick a graph Gn uniformly at random from all (labeled) perfect graphs on n vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that Gn satisfies it does not converge as n → ∞.