Polychromatic Coloring for Half-Planes

TL;DR

This paper establishes optimal bounds for polychromatic coloring of points and half-planes in the plane, improving previous bounds and providing new proofs for related geometric covering problems.

Contribution

It introduces tight bounds for coloring points so that large half-planes contain all colors, and for coloring half-planes to ensure coverage of points with multiple half-planes, advancing geometric coloring theory.

Findings

Optimal bound of 2k-1 for point coloring in half-planes.

Improved upper bound of 3k-2 for coloring half-planes.

New proof for small epsilon-nets in geometric range spaces.

Abstract

We prove that for every integer $k$, every finite set of points in the plane can be $k$-colored so that every half-plane that contains at least $2k-1$ points, also contains at least one point from every color class. We also show that the bound $2k-1$ is best possible. This improves the best previously known lower and upper bounds of $\frac{4}{3}k$ and $4k-1$ respectively. We also show that every finite set of half-planes can be $k$ colored so that if a point $p$ belongs to a subset $H_p$ of at least $3k-2$ of the half-planes then $H_p$ contains a half-plane from every color class. This improves the best previously known upper bound of $8k-3$. Another corollary of our first result is a new proof of the existence of small size $\eps$-nets for points in the plane with respect to half-planes.