We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored so that anymember ofF that contains at leastm points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then such an m exists. We prove this in the special case when F is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions
