On Mutually Avoiding Sets

Abstract

Two finite sets of points in the plane are called mutually avoiding if any straight line passing through two points of any one of these two sets does not intersect the convex hull of the other set. For any integer n, we construct a set of n points in general position in the plane which contains no pair of mutually avoiding sets of size more than \(\mathcal{O}(\sqrt{n})\) each. The given bound is tight up to a constant factor, since Aronov et al. [1] showed a polynomial-time algorithm for finding two mutually avoiding sets of size \(\Omega (\sqrt{n})\) each in any set of n points in general position in the plane.