Extremal Problems for Geometric Hypergraphs

A geometric hypergraph H is a collection of i -dimensional simplices, called hyperedges or, simply, edges, induced by some (i+1) -tuples of a vertex set V in general position in d -space. The topological structure of geometric graphs, i.e., the case d=2 , i=1 , has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and computational geometry. They include the k -set problem, proximity questions, bounding the number of incidences between points and lines, designing various efficient graph drawing algorithms, etc. In this paper, we make an attempt to generalize some of these tools to higher dimensions. We will mainly consider extremal problems of the following type. What is the largest number of edges (i -simplices) that a geometric hypergraph of n vertices can have without containing certain forbidden configurations? In particular, we discuss the special cases when the forbidden configurations are k intersecting edges, k pairwise intersecting edges, k crossing edges, k pairwise crossing edges, k edges that can be stabbed by an i -flat, etc. Some of our estimates are tight.

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

[2]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[3]  Endre Szemerédi,et al.  An upper bound on the number of planarK-sets , 1992, Discret. Comput. Geom..

[4]  Farhad Shahrokhi,et al.  Applications of the crossing number , 1994, SCG '94.

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

[6]  Yakov Shimeon Kupitz.  Extremal problems in combinatorial geometry , 1979 .

[7]  Endre Szemerédi,et al.  A Combinatorial Distinction Between the Euclidean and Projective Planes , 1983, Eur. J. Comb..

[8]  Zoltán Füredi,et al.  On the number of halving planes , 1989, SCG '89.

[9]  Tamal K. Dey.  On Counting Triangulations in D Dimensions , 1993, Comput. Geom..

[10]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..

[11]  J. Pach,et al.  Combinatorial geometry , 1995, Wiley-Interscience series in discrete mathematics and optimization.

[12]  N. Alon,et al.  Disjoint Simplices and Geometric Hypergraphs , 1989 .

[13]  János Pach,et al.  Some geometric applications of Dilworth’s theorem , 1994, Discret. Comput. Geom..

[14]  János Pach.  Notes on Geometric Graph Theory , 1990, Discrete and Computational Geometry.

[15]  Micha Sharir,et al.  Quasi-planar graphs have a linear number of edges , 1995, GD.

[16]  László A. Székely,et al.  Crossing Numbers and Hard Erdős Problems in Discrete Geometry , 1997, Combinatorics, Probability and Computing.

[17]  E. Szemerédi,et al.  Crossing-Free Subgraphs , 1982 .

[18]  János Pach,et al.  Ramsey-type results for geometric graphs , 1996, SCG '96.

[19]  Sinisa T. Vrecica,et al.  The Colored Tverberg's Problem and Complexes of Injective Functions , 1992, J. Comb. Theory A.

[20]  P. Erdös.  On extremal problems of graphs and generalized graphs , 1964 .

[21]  Leonidas J. Guibas,et al.  Combinatorial complexity bounds for arrangements of curves and spheres , 1990, Discret. Comput. Geom..