eBook ISBN:  9781470404468 
Product Code:  MEMO/179/845.E 
List Price:  $59.00 
MAA Member Price:  $53.10 
AMS Member Price:  $35.40 
eBook ISBN:  9781470404468 
Product Code:  MEMO/179/845.E 
List Price:  $59.00 
MAA Member Price:  $53.10 
AMS Member Price:  $35.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 179; 2006; 66 ppMSC: Primary 05
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

Chapters

1. Introduction

2. Outline of the proof

3. Tepees and constellations

4. Regularity

5. The core section (proof of Lemma 2.4)

6. Random graphs

7. Summary, further remarks, glossary


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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.

Chapters

1. Introduction

2. Outline of the proof

3. Tepees and constellations

4. Regularity

5. The core section (proof of Lemma 2.4)

6. Random graphs

7. Summary, further remarks, glossary