  Let $ES(n)$ be the minimal integer such that any set of $ES(n)$ points in the
plane in general position contains $n$ points in convex position. The problem
of estimating $ES(n)$ was first formulated by Erd\H{o}s and Szekeres, who
proved that $ES(n) \leq \binom{2n-4}{n-2}+1$. The current best upper bound,
$\lim\sup_{n \to \infty} \frac{ES(n)}{\binom{2n-5}{n-2}}\le \frac{29}{32}$, is
due to Vlachos. We improve this to $$\lim\sup_{n \to \infty}
\frac{ES(n)}{\binom{2n-5}{n-2}}\le \frac{7}{8}.$$
