Garment numbers of bi-colored point sets in the plane

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.