Empty rainbow triangles in k-colored point sets

Abstract

Let S be a set of n points in general position in the plane. Suppose that each point of S has been assigned one of k≥3 possible colors and that there is the same number, m, of points of each color class. This means n=km. A polygon with vertices on S is empty if it does not contain points of S in its interior; and it is rainbow if all its vertices have different colors. Let f(k,m) be the minimum number of empty rainbow triangles determined by S. In this paper we give tight asymptotic bounds for this function. Furthermore, we show that S may not determine an empty rainbow quadrilateral for some arbitrarily large values of k and m.