On a conjecture of Erd\H{o}s and Szekeres

TL;DR

This paper improves the upper bound on the Erd ext{"o}s-Szekeres function f(n), which determines the minimum number of points needed to guarantee a convex n-gon, showing it is asymptotically at most 29/32 of a certain binomial coefficient.

Contribution

The paper advances the understanding of f(n) by providing a tighter asymptotic upper bound, reducing the gap towards the conjectured optimal bound.

Findings

Improved the asymptotic upper bound of f(n) to 29/32 of a binomial coefficient.

Demonstrated that f(n) grows at most proportionally to (29/32) times the binomial coefficient as n increases.

Refined previous bounds, bringing us closer to the conjectured exact value of f(n).

Abstract

Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In a seminal paper published in 1935, Erd\H{o}s and Szekeres proved that f(n) exists and provided an upper bound. In 1961, they also proved a lower bound, which they conjectured is optimal. Their bounds are: $2^{n-2}+1 \leq f(n) \leq {2n - 4 \choose n-2}+1$. Since then, the upper bound has been improved by rougly a factor of 2, to $f(n) \leq {2n - 5 \choose n-2}+1$. In the current paper, we further improve the upper bound by proving that: $$ \limsup\limits_{n\rightarrow \infty} \frac{f(n)}{{2n-5 \choose n-2}} \leq \frac{29}{32}$$