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

Let $\mathbb{R}$ 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 $\widehat{c}=\widehat{c}(n)$ with $0<c<\widehat{c}<C$ such that for any $\text{eps}>0$, as $n$ tends to infinity $$Pr[G(n,(1-\text{eps})\widehat{c}\mathrm{/}\sqrt{n})\in \mathbb{R}]\to 0$$ and $$Pr[G(n,(1+\text{eps})\widehat{c}\mathrm{/}\sqrt{n})\in \mathbb{R}]\to 1.$$ 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.