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- W2778710285 abstract "In this paper, we study the embedding of a complete balanced d-partite d-uniform hypergraph with its nd vertices represented as points in general position in $${mathbb {R}}^d$$ and each hyperedge drawn as the convex hull of d corresponding vertices. We assume that the set of vertices is partitioned into d disjoint sets, each of size n, such that each vertex in a hyperedge is from a different set. Two hyperedges are said to be crossing if they are vertex disjoint and contain a common point in their relative interiors. Using Colored Tverberg theorem with restricted dimensions, we observe that such an embedding of a complete balanced d-partite d-uniform hypergraph with nd vertices contains $${varOmega }left( (8/3)^{d/2}right) {left( {n/2}right) ^d {left( (n-1)/2right) }^d}$$ crossing pairs of hyperedges for $$n ge 3$$ and sufficiently large d. Using Gale transform and Ham-Sandwich theorem, we improve this lower bound to $$ {varOmega }left( 2^{d}right) {left( {n/2}right) ^d{left( (n-1)/2 right) }^d}$$ for $$n ge 3$$ and sufficiently large d." @default.
- W2778710285 created "2018-01-05" @default.
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- W2778710285 date "2020-04-03" @default.
- W2778710285 modified "2023-10-12" @default.
- W2778710285 title "Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs" @default.
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- W2778710285 doi "https://doi.org/10.1007/s00373-020-02163-y" @default.
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