Canonical colourings in random graphs

Abstract: R\"odl and Ruci\'nski (1990) established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $\hat p_{K_\ell,r}(n)=n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_\ell$. We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation, Erd\H{o}s and Rado showed that canonically coloured copies of $K_\ell$ can be ensured in the deterministic setting. We transfer the Erd\H{o}s-Rado theorem to the random environment and show that both thresholds coincide for $\ell\geq 4$. As a consequence, the proof yields $K_{\ell+1}$-free graphs $G$ for which every edge colouring contains a canonically coloured $K_\ell$. The $0$-statement of the threshold is a direct consequence of the corresponding statement of the R\"odl-Ruci\'nski theorem and the main contribution is the $1$-statement. The proof of the $1$-statement employs the transference principle of Conlon and Gowers.