Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs

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 \({\varOmega }\left( (8/3)^{d/2}\right) {\left( {n/2}\right) ^d {\left( (n-1)/2\right) }^d}\) 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.