Let R(k) be the kth diagonal Ramsey number: that is, the smallest n for which every two-colouring of the edges of the complete graph on n vertices contains a monochromatic complete graph on k vertices. Since their introduction by Ramsey in the 1930s, these numbers have become a central topic in combinatorics and have inspired many beautiful advances in the intervening years, including the development of the probabilistic method and pseudo-random graphs.

In recent work with Marcelo Campos, Simon Griffiths and Rob Morris, the speaker showed that  R(k) < (4-c)^k, for some absolute constant c > 0, which was the first exponential improvement over the bound of Erdős and Szekeres, proved in 1935. In this talk I will discuss the proof and a further connection with a conjecture on random variables that take values in high dimensional space. If true, this conjecture has further implications for our understanding of the Ramsey numbers.