Two extensions of the Erd\H{o}s-Szekeres problem

TL;DR

This paper extends Suk's Erdős–Szekeres theorem to pseudoline arrangements, improves the error term, and applies these results to bound the size of convex families in the plane with specific intersection properties.

Contribution

The authors generalize the Erdős–Szekeres theorem to pseudoline arrangements and refine the bounds on convex configurations of convex bodies.

Findings

Generalization of Suk's theorem to pseudoline arrangements.

Improved upper bounds on functions c(n) and c'(n) for convex bodies.

Enhanced error term in the Erdős–Szekeres type bounds.

Abstract

According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at most $2^{n+O\left({n^{2/3}\log n}\right)}$ points. We strengthen this theorem in two ways. First, we show that the result generalizes to convexity structures induced by pseudoline arrangements. Second, we improve the error term. A family of $n$ convex bodies in the plane is said to be in convex position if the convex hull of the union of no $n-1$ of its members contains the remaining one. If any three members are in convex position, we say that the family is in general position. Combining our results with a theorem of Dobbins, Holmsen, and Hubard, we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes Toth and by Pach and Toth, respectively. Let $c(n)$ (and $c'(n)$) denote the smallest positive integer $N$ with the property that any family of $N$ pairwise disjoint convex bodies in general position (resp., $N$ convex bodies in general position, any pair of which share at most two boundary points) has an $n$-membered subfamily in convex position. We show that $c(n)\le c'(n)\leq 2^{n+O\left(\sqrt{n\log n}\right)}$.