Blocking Colored Point Sets

This paper studies problems related to visibility among points in the plane. A point xblocks two points v and w if x is in the interior of the line segment $$\overline{vw}$$ . 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 $$\{{n}_{1},{n}_{2},{n}_{3},{n}_{4}\}$$ 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.