Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of rb-index(S) over all k-colored point sets in general position; that is, every k-colored point set S has a perfect rainbow polygon with at most rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)≠k, and we prove that for k≥5, [Formula presented] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most [Formula presented] vertices can be computed in O(nlogn) time.