On the Evolution of Triangle-Free Graphs

Abstract

Let ${\cal T}(n,m)$ denote the set of all labelled triangle-free graphs with $n$ vertices and exactly $m$ edges. In this paper we give a short self-contained proof of the fact that there exists a constant $C>0$ such that, for all $m\geq Cn^{3/2}\sqrt{\log n}$, a graph chosen uniformly at random from ${\cal T}(n,m)$ is with probability $1-o(1)$ bipartite.