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NENADOV, RAJKO; STEGER, ANGELIKA
Combinatorics, probability & computing, 01/2016, Letnik: 25, Številka: 1Journal Article
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 $$ \begin{equation*} \PrG_{n,p} \rightarrow (F)_r^e = \begin{cases} 1-o(1) &p\ge Cn^{-1/m_2(F)},\\ o(1) &p\le cn^{-1/m_2(F)}, \end{cases} \end{equation*} $$ 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.
