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

Released on by Ehud Friedgut
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  • A Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring Book Detail

  • Author : Ehud Friedgut
  • Release Date : 2006
  • Publisher : American Mathematical Soc.
  • Genre : Mathematics
  • Pages : 80
  • ISBN 13 : 0821838253
  • File Size : 9,9 MB
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A Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring by Ehud Friedgut PDF Summary

Book 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 the authors establish a sharp threshold for random graphs with this property. Let $G(n, p)$ be the random graph on $n$ vertices with edge probability $p$. The authors 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 setti

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