We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set of points in the plane S⊂ R2 can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering 2-coloring points with respect to homothets of a fixed parallelogram.
Bibliographical noteFunding Information:
E. Ackerman’s research partially supported by ERC Advanced Research Grant No. 267165 (DISCONV). B. Keszegh’s research supported by the National Research, Development and Innovation Office – NKFIH under the Grant PD 108406 and K 116769 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. M. Vizer’s research supported by the National Research, Development and Innovation Office – NKFIH under the Grant SNN 116095. We thank the reviewers for reading our paper and for their valuable remarks. In particular, suggestions of one of reviewers helped to simplify some basic lemmas and improve the constant in our main theorem.
© 2017, Springer Science+Business Media New York.
- Geometric hypergraph coloring
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics