Marcelo Campos

An exponential improvement for diagonal Ramsey

The Ramsey number R(k) is the minimum natural number n such that every red-blue colouring of the edges of the complete graph K_n on n vertices contains a monochromatic copy of K_k. In this talk I will present a recent result which shows that R(k) \le (4 - \eps)^k for some constant \eps > 0. This is the first exponential improvement on the upper bound of Erdos and Szekeres, proved in 1935. Joint work with Simon Griffiths, Robert Morris and Julian Sahasrabudhe.