Polychromatic Coloring for Half-Planes

  • Shakhar Smorodinsky
  • Yelena Yuditsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)


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. As a corollary, 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 4k − 3 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.


Geometric Hypergraphs Discrete Geometry Polychromatic Coloring ε-Nets 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Shakhar Smorodinsky
    • 1
  • Yelena Yuditsky
    • 1
  1. 1.Ben-Gurion UniversityBe’er ShevaIsrael

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