Upper Bounds for Multicolour Ramsey Numbers

The r-colour Ramsey number Rr(k) is the minimum n∈ℕ such that every r-colouring of the edges of the complete graph Kn on n vertices contains a monochromatic copy of Kk. We prove, for each fixed r≥2, that

Rr(k)≤e−δkrrk

for some constant δ=δ(r)>0 and all sufficiently large k∈ℕ. For each r≥3, this is the first exponential improvement over the upper bound of Erdős and Szekeres from 1935. In the case r=2, it gives a different (and significantly shorter) proof of a recent result of Campos, Griffiths, Morris and Sahasrabudhe.  This is based on joint work with Paul Balister, Bela Bollobas, Marcelo Campos, Simon Griffiths, Eoin Hurley, Robert Morris and Julian Sahasrabudhe.