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

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*Ehud Friedgut; Vojtech Rödl; Andrzej Ruciński; Prasad Tetali*

Let \(\mathcal{R}\) 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 \(\widehat c=\widehat c(n)=\Theta(1)\) such that for any \(\varepsilon > 0\), as \(n\) tends to infinity, \(Pr\left[G(n,(1-\varepsilon)\widehat c/\sqrt{n}) \in \mathcal{R} \right] \rightarrow 0\) and \(Pr \left[ G(n,(1+\varepsilon)\widehat c/\sqrt{n}) \in \mathcal{R}\ \right] \rightarrow 1\). 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.