A Sharp Threshold for Random Graphs With a Monochromatic Triangle in Every Edge Coloring (Memoirs of the American Mathematical Society) - Softcover

Rodl, Vojtech; Rucinski, Andrzej

Softcover

ISBN 10: 0821838253 ISBN 13: 9780821838259

Publisher: Amer Mathematical Society, 2005

Synopsis

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 Szemerédi's Regularity Lemma to a certain hypergraph setting.

Edition details

• Publisher: Amer Mathematical Society
• Publication date: 2005
• Language: English
• ISBN 10: 0821838253
• ISBN 13: 9780821838259
• Binding: Paperback
• Number of pages: 66
• Editor: Friedgut Ehud