Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring 1

Vetenskap & teknik

Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring

Ehud Friedgut • Vojtech Rodl • Andrzej Rucinski • Prasad Tetali

Pocket

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2005

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.

Författare: Ehud Friedgut, Vojtech Rodl, Andrzej Rucinski, Prasad Tetali
Format: Pocket/Paperback
ISBN: 9780821838259
Språk: Engelska
Utgivningsdatum: 2005-12-30
Förlag: American Mathematical Society