Balanced Line for a 3-Colored Point Set in the Plane

Abstract

In this note we prove the following theorem. For any three sets of points in the plane, each of $n\ge 2$ points such that any three points (from the union of three sets) are not collinear and the convex hull of $3n$ points is monochromatic, there exists an integer $k\in\{1,2,\dots,n-1\}$ and an open half-plane containing exactly $k$ points from each set.