An Improved Upper Bound for the Erdős–Szekeres Conjecture

Abstract

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, Erdős and Szekeres proved that ES\((n) \le {2n-4 \atopwithdelims ()n-2}+1\). In 1961, they obtained the lower bound \(2^{n-2}+1 \le \mathrm{ES}(n)\), which they conjectured to be optimal. In this paper, we prove that $$\begin{aligned} \mathrm{ES}(n) \le \Big (\begin{array}{c}{2n-5} \\ {n-2}\end{array}\Big ) -\Big (\begin{array}{c}{2n-8} \\ {n-3}+2\end{array}\Big ) \approx \frac{7}{16} \Big (\begin{array}{c}{2n-4}\\ {n-2}\end{array}\Big ). \end{aligned}$$