Point Sets with Many k-Sets

For any n , k , n\geq 2k>0 , we construct a set of n points in the plane with $ne^{\Omega({\sqrt{\log k}})}$ k -sets. This improves the bounds of Erdős, Lovász, et al. As a consequence, we also improve the lower bound for the number of halving hyperplanes in higher dimensions.

[1]  János Pach,et al.  Combinatorial Geometry , 2012 .

[2]  Leonidas J. Guibas,et al.  Points and triangles in the plane and halving planes in space , 1991, Discret. Comput. Geom..

[3]  Emo Welzl.  More onk-sets of finite sets in the plane , 1986, Discret. Comput. Geom..

[4]  R. Pollack,et al.  Allowable Sequences and Order Types in Discrete and Computational Geometry , 1993 .

[5]  Richard Pollack,et al.  On the Number of k-Subsets of a Set of n Points in the Plane , 1984, J. Comb. Theory, Ser. A.

[6]  Herbert Edelsbrunner,et al.  Constructing Belts in Two-Dimensional Arrangements with Applications , 1986, SIAM J. Comput..

[7]  Herbert Edelsbrunner,et al.  On the Number of Line Separations of a Finite Set in the Plane , 1985, J. Comb. Theory, Ser. A.

[8]  Micha Sharir,et al.  An Improved Bound for k-Sets in Three Dimensions , 2000, SCG '00.

[9]  P. Erdös,et al.  Dissection Graphs of Planar Point Sets , 1973 .

[10]  Tamal K. Dey,et al.  Improved Bounds for Planar k -Sets and Related Problems , 1998, Discret. Comput. Geom..

[11]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[12]  Herbert Edelsbrunner,et al.  Cutting dense point sets in half , 1997, Discret. Comput. Geom..

[13]  David Eppstein.  Improved Bounds for Intersecting Triangles and Halving Planes , 1993, J. Comb. Theory, Ser. A.

[14]  H. Edelsbrunner,et al.  Counting triangle crossings and halving planes , 1994, Discret. Comput. Geom..

[15]  David Eppstein.  Sets of points with many halving lines , 1992 .