Asymmetric Ramsey properties of random graphs involving cliques and cycles

Abstract

We prove that for every $\ell,r \geq 3$, there exists $c>0$ such that for $p \leq cn^{-1/m_2(K_r,C_{\ell})}$, with high probability there is a 2-edge-colouring of the random graph $\gnp$ with no monochromatic copy of $K_r$ of the first colour and no monochromatic copy of $C_\ell$ of the second colour. This is a progress on a conjecture of Kohayakawa and Kreuter.