Erd\H{o}s-Szekeres without induction

TL;DR

This paper improves the upper bound on the minimal number of points needed in the plane to guarantee a convex n-gon, refining previous estimates without using induction.

Contribution

The authors provide a new upper bound on $ES(n)$, reducing the limit supremum ratio from 29/32 to 7/8, advancing the understanding of convex configurations.

Findings

Improved upper bound on $ES(n)$ ratio to 7/8

Refined estimation without induction methods

Progress towards tight bounds in convex point set problems

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 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 $$\lim\sup_{n \to \infty} \frac{ES(n)}{\binom{2n-5}{n-2}}\le \frac{7}{8}.$$