An improved upper bound for the Erd\H{o}s-Szekeres conjecture

TL;DR

This paper improves the upper bound for the Erd ext{"o}s-Szekeres conjecture, narrowing the gap between known bounds for the minimum number of points needed to guarantee a convex n-gon in any point set.

Contribution

The authors establish a tighter upper bound for ES(n), advancing the understanding of the minimal point set size required for convex polygons.

Findings

New upper bound: ES(n)  {2n-5  n-2} - {2n-8  n-3} + 2

Approximate ratio:  rac{7}{16} of previous upper bound

Progress towards resolving the Erd ext{"o}s-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, 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 $$ES(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2 \approx \frac{7}{16} {2n-4 \choose n-2}.$$