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A Short Proof of the Random Ramsey Theorem

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  22 December 2014

RAJKO NENADOV  and
ANGELIKA STEGER
RAJKO NENADOV
Affiliation:
Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland (e-mail: rnenadov@inf.ethz.ch, steger@inf.ethz.ch)
ANGELIKA STEGER
Affiliation:
Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland (e-mail: rnenadov@inf.ethz.ch, steger@inf.ethz.ch)
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Abstract

In this paper we give a short proof of the Random Ramsey Theorem of Rödl and Ruciński: for any graph F which contains a cycle and r ≥ 2, there exist constants c, C > 0 such that

$$\begin{equation*}\Pr[G_{n,p} \rightarrow (F)_r^e] =\begin{cases}1-o(1) &p\ge Cn^{-1/m_2(F)},\\o(1) &p\le cn^{-1/m_2(F)},\end{cases}\end{equation*}$$
where
$$\begin{equation*}m_2(F) = \max_{J\subseteq F, v_J\ge 2} \frac{e_J-1}{v_J-2}.\end{equation*}$$
The proof of the 1-statement is based on the recent beautiful hypergraph container theorems by Saxton and Thomason, and Balogh, Morris and Samotij. The proof of the 0-statement is elementary.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05D10: Ramsey theory

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 1: Oberwolfach Special Issue Part 2 , January 2016 , pp. 130 - 144
Copyright
Copyright © Cambridge University Press 2014 

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