On the Erdos-Szekeres convex polygon problem

TL;DR

This paper advances the understanding of the Erdos-Szekeres convex polygon problem by nearly proving the conjecture that the minimum number of points needed to guarantee an n-point convex polygon grows exponentially with n.

Contribution

The paper provides a near-complete proof that the minimal number of points in general position to ensure an n-point convex polygon is asymptotically 2^{n}, nearly settling the longstanding conjecture.

Findings

Established that ES(n) = 2^{n + o(n)}

Improved bounds on the minimal point set size for convex polygons

Nearly proved the Erdos-Szekeres conjecture

Abstract

Let $ES(n)$ be the smallest integer such that any set of $ES(n)$ points in the plane in general position contains $n$ points in convex position. In their seminal 1935 paper, Erdos and Szekeres showed that $ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}$. In 1960, they showed that $ES(n) \geq 2^{n-2} + 1$ and conjectured this to be optimal. In this paper, we nearly settle the Erdos-Szekeres conjecture by showing that $ES(n) =2^{n +o(n)}$.