## 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 43k 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 ε-nets for points in the plane with respect to half-planes.

Original language | English |
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Pages (from-to) | 146-154 |

Number of pages | 9 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 119 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2012 |

## Keywords

- Cover decomposable
- Discrete geometry
- Epsilon nets
- Polychromatic coloring

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics