Mathias Schacht, Canonical colourings in random graphs

Rödl and Ruciński established Ramsey’s theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $n^{-\frac{2}{\ell+1}}$ 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 Erdős and Rado showed that canonically coloured copies of $K_\ell$ can be ensured in the deterministic setting.
We transfer the Erdős-Rado theorem to the random environment and show that for $\ell\geq 4$ both thresholds coincide. As a consequence the proof yields $K_{\ell+1}$-free graphs $G$ for which every edge colouring yields a canonically coloured $K_\ell$.

This is joint work with Nina Kamčev.