A density version of the Hales–Jewett theorem for k=3
Discrete Mathematics
Some new bounds for Epsilon-nets
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
How to net a lot with little: small &egr;-nets for disks and halfspaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Almost tight bounds for &egr;-nets
Discrete & Computational Geometry
Reporting points in halfspaces
Computational Geometry: Theory and Applications
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
Proceedings of the twenty-fourth annual symposium on Computational geometry
Efficient Colored Orthogonal Range Counting
SIAM Journal on Computing
Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles
Random Structures & Algorithms - Proceedings of the Thirteenth International Conference “Random Structures and Algorithms” held May 28–June 1, 2007, Tel Aviv, Israel
Lower bounds for weak epsilon-nets and stair-convexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
Epsilon nets and union complexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
Hitting sets when the VC-dimension is small
Information Processing Letters
Decomposing Coverings and the Planar Sensor Cover Problem
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Notes: Coloring axis-parallel rectangles
Journal of Combinatorial Theory Series A
A note about weak ε-nets for axis-parallel boxes in d-space
Information Processing Letters
A Non-linear Lower Bound for Planar Epsilon-Nets
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Cover-decomposition and polychromatic numbers
ESA'11 Proceedings of the 19th European conference on Algorithms
Hardness of discrepancy computation and ε-net verification in high dimension
Journal of Complexity
The class cover problem with boxes
Computational Geometry: Theory and Applications
Piercing quasi-rectangles-On a problem of Danzer and Rogers
Journal of Combinatorial Theory Series A
Optimal area-sensitive bounds for polytope approximation
Proceedings of the twenty-eighth annual symposium on Computational geometry
Small-size relative (p,ε)-approximations for well-behaved range spaces
Proceedings of the twenty-ninth annual symposium on Computational geometry
Coloring hypergraphs induced by dynamic point sets and bottomless rectangles
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
Exact algorithms and APX-hardness results for geometric packing and covering problems
Computational Geometry: Theory and Applications
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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size O(1/ε log 1/ε). Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ε-nets is superlinear in 1/ε, were found by Alon (2010). In his examples, every ε-net is of size Ω(1/εg(1/ε)), where g is an extremely slowly growing function, related to the inverse Ackermann function. We show that there exist geometrically defined range spaces, already of VC-dimension 2, in which the size of the smallest ε-nets is Ω(1/ε log 1/ε). We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is Ω(1/ε log log 1/ε). By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.