Blocking colored point sets

This paper studies problems related to visibility among points in the plane. A point x blocks two points v and w if x is in the interior of the line segment vw¯. A set of points P is k-blocked if each point in P is assigned one of k colors, such that distinct points v, w ∈ P are assigned the same color if and only if some other point in P blocks v and w. The focus of this paper is the conjecture that each k-blocked set has bounded size (as a function of k). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterize all sets {n1,n2,n3,n4} such that some 4-blocked set has exactly n_i points in the ith color class. Among other results, for infinitely many values of k, we construct k-blocked sets with k^1.79...points.