Canonical colourings in random graphs

TL;DR

This paper extends classical results on monochromatic and canonical colourings in random graphs, establishing thresholds for the appearance of canonically coloured cliques in graphs with many colours, using advanced probabilistic methods.

Contribution

It transfers the Erdős-Rado theorem to the random graph setting and determines the threshold for canonical colourings in this environment.

Findings

Threshold for canonical colourings matches the monochromatic case for ll.

Established the existence of ll+1-free graphs with canonical ll-cliques in random graphs.

Applied the transference principle of Conlon and Gowers to prove the main results.

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.