# A Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring (Memoirs of the American Mathematical Society No. 179 Illustr

By: Andrzej Rucinski (author), Prasad Tetali (author), Vojtech Rodl (author), Ehud Friedgut (author)Paperback

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### Description

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 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, $Pr\left[G(n,(1-\varepsilon)\widehat c/\sqrt{n}) \in \cal{R} \right] \rightarrow 0$ 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 Szemeredi's Regularity Lemma to a certain hypergraph setting.

### 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.

### Product Details

• publication date: 15/12/2005
• ISBN13: 9780821838259
• Format: Paperback
• Number Of Pages: 66
• ID: 9780821838259
• weight: 167
• ISBN10: 0821838253
• edition: Illustrated edition

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