Abstract
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.
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Research supported in part by DST-SR-OY-E-06-95 grant, India
Research supported by NSF grant CCR-94-24398, PSC-CUNY Research Award 663472, and OTKA-4269.
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References
M. Ajtai, V. Chvátal, M. M. Newborn, and E. Szemerédi. Crossing-free subgraphs. Ann. Discrete Math. 12 (1982), 9–12.
P. K. Agarwal, B. Aronov, J. Pach, R. Pollack, and M. Sharir. Quasi-planar graphs have a linear number of edges. In: Graph Drawing '95, Lecture Notes in Computer Science 1027, Springer-Verlag, Berlin, 1996, 1–7. Also in: Combinatorica (to appear).
J. Akiyama and N. Alon. Disjoint simplices and geometric hypergraphs. Combinatorial Mathematics (G. S. Bloom et al., eds.), Annals of the New York Academy of Sciences 555 (1989), 1–3.
B. Aronov, B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, and R. Wenger. Points and triangles in the plane and halving planes in space. Discrete and Computational Geometry 6 (1991), 435–442.
I. Bárány, Z. Füredi, and L. Lovász. On the number of halving planes. Combinatorica 10 (1990), 175–183.
K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl. Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom. 5 (1990), 99–160.
T. K. Dey. On counting triangulations in d dimensions. Computational geometry: Theory and Applications. 3 (1993), 315–325.
T. K. Dey and H. Edelsbrunner. Counting triangle crossings and halving planes. 9th Sympos. Comput. Geom. (1993), 270–273. Also: Discrete and Computational Geometry 12 (1994), 281–289.
P. Erdős. On extremal problems on graphs and generalized graphs. Israel J. Math. 2 (1964), 183–190.
G. Károlyi, J. Pach, and G. Tóth. Ramsey-type results for geometric graphs. 12th Sympos. Comput. Geom. (1996). Also in: Discrete and Computational Geometry (to appear).
Y. Kupitz. Extremal Problems in Combinatorial Geometry, Aarhus University Lecture Notes Series 53, Aarhus University, Denmark, 1979.
F. T. Leighton. Complexity Issues in VLSI, Foundations of Computing Series, MIT Press, Cambridge, Mass., 1983.
R. Lipton and R. Tarjan. A separator theorem for planar graphs. SIAM J. Applied Mathematics 36 (1979), 177–189.
L. Lovász. On the number of halving lines. Annales Universitatis Scientarium Budapest, Eötvös, Sectio Mathematica 14 (1971), 107–108.
J. Pach. Notes on geometric graph theory. DIMACS Ser. Discr. Math. and Theoret. Comput. Sc. 6 (1991), 273–285.
J. Pach and P. K. Agarwal. Combinatorial Geometry, Wiley, New York, 1995.
J. Pach and M. Sharir. On the number of incidences between points and curves (to appear).
J. Pach, F. Shahrokhi, and M. Szegedy. Applications of crossing numbers. 10th ACM Sympos. Comput. Geom. (1994), 198–202.
J. Pach, W. Steiger, and M. Szemerédi. An upper bound on the number of planar k-sets. Discrete and Computational Geometry 7 (1992), 109–123.
J. Pach and J. Törőcsik. Some geometric applications of Dilworth's theorem. 9th Sympos. Comput. Geom. (1993), 264–269. Also in: Discrete and Computational Geometry 12 (1994), 1–7.
L. A. Székely, Crossing numbers and hard Erdős problems in discrete geometry. Combinatorics, Probability, and Computing, (to appear).
E. Szemerédi and W. T. Trotter. Extremal problems in discrete geometry. Combinatorica 3 (1983), 381–392.
E. Szemerédi and W. T. Trotter. A combinatorial distinction between the Euclidean and projective planes. European J. Combinatorics 4 (1983), 385–394.
R. Tamassia and I. Tollis (eds.). Graph Drawing, Lecture Notes in Computer Science 894, Springer-Verlag, Berlin, 1995.
S. Vrećica and R. Živaljević. New cases of the colored Tverberg's theorem. In: Jerusalem Combinatorics '93, Contemp. Math. 178, Amer. Math. Soc., Providence, 1994, 325–334.
R. Živaljević and S. Vrećica. The colored Tverberg's problem and complexes of injective functions. J. Combin. Theory Ser. A 61 (1992), 309–318.
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Dey, T.K., Pach, J. (1996). Extremal problems for geometric hypergraphs. In: Asano, T., Igarashi, Y., Nagamochi, H., Miyano, S., Suri, S. (eds) Algorithms and Computation. ISAAC 1996. Lecture Notes in Computer Science, vol 1178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0009486
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DOI: https://doi.org/10.1007/BFb0009486
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