Garment numbers of bi-colored point sets in the plane

TL;DR

This paper investigates monochromatic geometric structures in bichromatic point sets in the plane, providing bounds on the minimum number of points needed to guarantee their existence, thus advancing understanding of geometric combinatorics.

Contribution

It introduces new bounds on the size of bichromatic point sets needed to ensure certain monochromatic geometric configurations, addressing open problems in the field.

Findings

Improved lower bounds on the minimum size for guaranteed monochromatic structures

Enhanced upper bounds for specific monochromatic geometric configurations

Progress towards resolving open problems on convex empty quadrilaterals

Abstract

We consider colored variants of a class of geometric-combinatorial questions on $k$-gons and empty $k$-gons that have been started around 1935 by Erd\H{o}s and Szekeres. In our setting we have $n$ points in general position in the plane, each one colored either red or blue. A structure on $k$ points is a geometric graph where the edges are spanned by (some of) these points and is called monochromatic if all $k$ points have the same color. Already for $k=4$ there exist interesting open problems. Most prominently, it is still open whether for any sufficiently large bichromatic set there always exists a convex empty, monochromatic quadrilateral. In order to shed more light on the underlying geometry we study the existence of five different monochromatic structures that all use exactly 4 points of a bichromatic point set. We provide several improved lower and upper bounds on the smallest $n$ such that every bichromatic set of at least $n$ points contains (some of) those monochromatic structures.