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A Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring
Ehud Friedgut, Hebrew University, Jersulem, Israel, Vojtech Rödl, Adam Mickiewicz University, Poznán, Poland, Andrzej Ruciński, Georgia Institute of Technology, Atlanta, GA, and Prasad Tetali
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Memoirs of the American Mathematical Society
2006; 66 pp; softcover
Volume: 179
ISBN-10: 0-8218-3825-3
ISBN-13: 978-0-8218-3825-9
List Price: US$59 Individual Members: US$35.40
Institutional Members: US\$47.20
Order Code: MEMO/179/845

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.