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 (Compos Math 2: 463–470, 1935), who proved that $ES(n) \le \left( {\begin{array}{c}2n-4\\ n-2\end{array}}\right) +1$. The current best upper bound, $\lim \sup _{n \rightarrow \infty } \tfrac{ES(n)}{\left( {\begin{array}{c}2n-5\\ n-2\end{array}}\right) }\le \tfrac{29}{32}$, is due to Vlachos (On a conjecture of Erdős and Szekeres, 2015). We improve this to

$$\begin{aligned} \lim \sup _{n \rightarrow \infty } \tfrac{ES(n)}{\left( {\begin{array}{c}2n-5\\ n-2\end{array}}\right) }\le \tfrac{7}{8}. \end{aligned}$$