Some new bounds for Epsilon-nets
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Almost tight bounds for &egr;-nets
Discrete & Computational Geometry
WG '88 Proceedings of the 14th International Workshop on Graph-Theoretic Concepts in Computer Science
Restricted strip covering and the sensor cover problem
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Multiple Coverings of the Plane with Triangles
Discrete & Computational Geometry
Decomposition of multiple coverings into many parts
Computational Geometry: Theory and Applications
Decomposition of multiple coverings into more parts
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Coloring Geometric Range Spaces
Discrete & Computational Geometry
Convex Polygons are Cover-Decomposable
Discrete & Computational Geometry
CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
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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 Hp of at least 4k−3 of the half-planes then Hp 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.