Ebook Info
- Published: 2005
- Number of pages: 66 pages
- Format: PDF
- File Size: 9.05 MB
- Authors: and Prasad Tetali Ehud Friedgut, Vojtech Rödl, Andrzej Rucinski
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, $Prleft[G(n,(1-varepsilon)widehat c/sqrt{n}) in cal{R} right] rightarrow 0$ 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.
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