A Short Proof of the Random Ramsey Theorem

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 P[Gn,p → (F )r] = { 1− o(1), p ≥ Cn−1/m2(F ) o(1), p ≤ cn−1/m2(F , where m2(F ) = maxJ⊆F,vJ≥2 eJ−1 vJ−2 . The proof of the 1-statement is based on the recent beautiful hypergraph container theorems by Saxton/Thomason and Balogh/Morris/Samotij. The proof of the 0-statement is elementary.

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