  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.
