Making Triangles Colorful

TL;DR

This paper proves a polynomial bound on coloring points in the plane so that large homothetic triangles contain all colors, advancing understanding of range spaces induced by polygons.

Contribution

It establishes the first polynomial bound for coloring points with respect to homothetic polygons, improving over previous doubly exponential bounds for octants.

Findings

Existence of a k-coloring ensuring all colors in large homothetic triangles

Polynomial bound of ck^8 points for the coloring property

First such bound for polygon-induced range spaces

Abstract

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 ck^8 points of P, for some constant c, 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 R^3, but is doubly exponential.