The Ramsey number $R(k)$ is the minimum $n \in \mathbb{ N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erd\H{ o}s and Szekeres, proved in 1935.
Categories: math.CO