  Let $ES(n)$ denote the minimum natural number such that every set of $ES(n)$
points in general position in the plane contains $n$ points in convex position.
In 1935, Erd\H{o}s and Szekeres proved that $ES(n) \le {2n-4 \choose n-2}+1$.
In 1961, they obtained the lower bound $2^{n-2}+1 \le ES(n)$, which they
conjectured to be optimal. In this paper, we prove that $$ES(n) \le {2n-5
\choose n-2}-{2n-8 \choose n-3}+2 \approx \frac{7}{16} {2n-4 \choose n-2}.$$

Combinatorics