Empty Rainbow Triangles in k-colored Point Sets

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, so n = km. A triangle 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 show that
f(k, m) = Θ(k 3). Furthermore we give a construction which does not contain an empty rainbow quadrilateral.