Canonical colourings in random graphs

Abstract

Rödl and Ruciński (J. Amer. Math. Soc. 8 (1995), 917–942) established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell ⩾2$ they proved that ${\hat{p}}_{K_{\ell },r}\left(n\right)=n^{-\frac{2}{\ell +1}}$ 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 context, Erdős and Rado (J. Lond. Math. Soc. 25 (1950), 249–255) proved that any edge-colouring of a sufficiently large complete graph contains one of four canonical colourings of $K_{\ell }$, a monochromatic, or rainbow, or min or max colouring; a min-colouring of $K_{\ell }$ is a colouring in which two edges have the same colour if and only if they have the same minimal vertex. We transfer the Erdős–Rado theorem to the random graph $G(n,p)$ and show that both thresholds coincide when $\ell ⩾4$. 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 Rödl–Ruciński theorem and the main contribution is the 1-statement. The proof of the 1-statement employs the transference principle of Conlon and Gowers (Ann. of Math. (2) 184 (2016), 367–454).