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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

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
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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
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