The first order convergence law fails for random perfect graphs

TL;DR

This paper demonstrates that the first order convergence law does not hold for random perfect graphs by providing an example of a property whose probability of satisfaction does not converge as the number of vertices grows.

Contribution

It shows the failure of the first order convergence law for random perfect graphs, highlighting limitations in logical convergence properties for this class.

Findings

Existence of a first order property with non-converging probability

Counterexample to the first order convergence law in perfect graphs

Implication for logical properties in random graph models

Abstract

We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) 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 $G_n$ satisfies it does not converge as $n\to\infty$.