Small weak epsilon nets

Given a set P of points in the plane, a set of points Q is a weak ε-net with respect to a family of sets S (e.g., rectangles, disks, or convex sets) if every set of S containing ε|P | points contains a point of Q. In this paper, we determine bounds on εi , the Preprint submitted to Elsevier Science 10 February 2008 smallest epsilon that can be guaranteed for any P when |Q| = i, for small values of i.