The Erd\H{o}s-Szekeres problem for non-crossing convex sets

TL;DR

This paper establishes an equivalence between two conjectures related to convex bodies and point sets, leading to improved bounds on the Erdős-Szekeres theorem for non-crossing convex sets and their arrangements.

Contribution

It introduces a novel equivalence between conjectures and improves existing bounds for Erdős-Szekeres type problems involving non-crossing convex bodies.

Findings

Improved upper bounds on Erdős-Szekeres theorem for disjoint convex bodies.

Enhanced bounds for non-crossing convex bodies in the Erdős-Szekeres context.

Generalization of the partitioned Erdős-Szekeres theorem for arrangements of non-crossing convex bodies.

Abstract

We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and T\'{o}th on the Erd\H{o}s-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk, on the Erd\H{o}s-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erd\H{os}-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erd\H{o}s-Szekeres theorem of P\'{o}r and Valtr to arrangements of non-crossing convex bodies.