Ramsey numbers for empty convex polygons

We study a geometric Ramsey type problem where the vertices of the complete graph Kn are placed on a set S of n points in general position in the plane, and edges are drawn as straight-line segments. We dene the empty convex polygon Ramsey number REC(k;k) as the smallest number n such that for every set S of n points and for every two-coloring of the edges of Kn drawn on S, at least one color class contains an empty convex k-gon. A polygon is empty if it contains no points from S in its interior. We prove 17 REC(3; 3) 463 and 57 REC(4; 4): Further, there are three-colorings of the edges of Kn (drawn on a set S) without empty monochromatic triangles. A related Ramsey number for islands in point sets is also studied.