Memoirs of the American Mathematical Society 2006; 66 pp; softcover Volume: 179 ISBN10: 0821838253 ISBN13: 9780821838259 List Price: US$56 Individual Members: US$33.60 Institutional Members: US$44.80 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. Table of Contents  Introduction
 Outline of the proof
 Tepees and constellations
 Regularity
 The core section (Proof of Lemma 2.4)
 Random graphs
 Summaryt, further remarks, glossary
 Bibliography
