A Sharp Threshold for Random Graphs With a Monochromatic Triangle in Every Edge Coloring

Let $\cal{R}$ 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 the authors establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. The authors prove that there exists a function $\widehat c=\widehat c(n)=\Theta(1)$ such that for any $\varepsilon > 0$, as $n$ tends to infinity,
$Pr\left[G(n,(1-\varepsilon)\widehat c/\sqrt{n}) \in \cal{R} \right] \rightarrow 0$ and $Pr \left[ G(n,(1+\varepsilon)\widehat c/\sqrt{n}) \in \cal{R}\ \right] \rightarrow 1.$ A crucial tool that is used in the proof and is of independent interest is a generalization of Szemerédi's Regularity Lemma to a
certain hypergraph setting.