# 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, 40⌊ (k-1)/2 ⌋ -8/19 ≤rb-index(k)≤ 10 ⌊k/7⌋ + 11. Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most 10 ⌊k/7⌋ + 11 vertices can be computed in O(nlog n) time.

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