Erdős-Szekeres without induction

Abstract

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 Erdős and Szekeres, who proved that $ES(n)\le \left(\genfrac{}{}{0ex}{}{2n-4}{n-2}\right)+1$. The current best upper bound, $\lim {\sup }_{n\to \infty }\frac{ES(n)}{\left(\genfrac{}{}{0ex}{}{2n-5}{n-2}\right)}\le \frac{29}{32}$, is due to Vlachos. We improve this to $$\lim \underset{n\to \infty }{\sup }\frac{ES(n)}{\left(\genfrac{}{}{0ex}{}{2n-5}{n-2}\right)}\le \frac{7}{8}.$$