making triangles colorful

We prove that for any point set P in the plane, a triangle T, and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least 144k⁸ points of P contains at least one of each color. This is the first polynomial bound for range spaces induced by homothetic polygons.The only previously known bound for this problem applies to the more general case of octants in ℝ³, but is doubly exponential.