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A Short Proof of the Random Ramsey Theorem

Published online by Cambridge University Press:  22 December 2014

Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland (e-mail:,
Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland (e-mail:,


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

$$ \begin{equation*} \Pr[G_{n,p} \rightarrow (F)_r^e] = \begin{cases} 1-o(1) &p\ge Cn^{-1/m_2(F)},\\ o(1) &p\le cn^{-1/m_2(F)}, \end{cases} \end{equation*} $$
$$ \begin{equation*} m_2(F) = \max_{J\subseteq F, v_J\ge 2} \frac{e_J-1}{v_J-2}. \end{equation*} $$
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

Copyright © Cambridge University Press 2014 

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