Rob Morris, An exponential improvement for diagonal Ramsey

The Ramsey number $R(k)$ is the minimum n such that every red-blue colouring of the edges of the complete graph on n vertices contains a monochromatic copy of $K_k$. It has been known since the work of Erdős and Szekeres in 1935, and Erdős in 1947, that $2^{k/2}<R(k)<4^k$, but in the decades since the only improvements have been by lower order terms. In this talk I will sketch the proof of a very recent result, which improves the upper bound of Erdős and Szekeres by a (small) exponential factor.

Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.