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.
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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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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