A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

Abstract

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\hat c=\hat c(n)$ with $0<c<\hat c<C$ such that for any $\eps > 0$, as $n$ tends to infinity $$Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0$$ and $$Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1.$$ A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.