On Visibility and Blockers

This expository paper discusses some conjectures related to visibility and blockers for sets of points in the plane

An algorithmic proof of the Motzkin-Rabin theorem on monochrome lines

We present a new proof of the following theorem originally due to Motzkin and Rabin (see [9, 1, 6, 7]). Motzkin-Rabin Theorem. Let S be a finite noncollinear set of points in the plane, each colored red or blue. Then there exists a line l passing through at least two points of S, all points of S on l being of the same color. We say that a set of points S is two-colored if each point in S is assigned one of the colors red or blue. A line passing through at least two points of S with all points of S on the line assigned the same color is called a monochrome line. It makes no di erence whether the plane in the theorem is the Euclidean or the projective plane, since if we are in the projective plane we can always find a line disjoint from the finite set S, project it to infinity, and then the set S can be considered to be in the Euclidean plane. There are two essentially different proofs of the Motzkin-Rabin theorem in the literature, both proving the projective dual of the theore..