Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons

Abstract

We consider a variation of a problem stated by Erdős and Szekeres in 1935 about the existence of a number $f^{\mathrm{ES}}(k)$ such that any set S of at least $f^{\mathrm{ES}}(k)$ points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any sufficiently large bichromatic set of points in $\mathbb{R}^2$ in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).