  We study whether for a given planar family F there is an m such that any
finite set of points can be 3-colored such that any member of F that contains
at least m 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.
