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

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 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 ( (8/3)^d/2) ( n/2) ^d ( (n-1)/2) ^d crossing pairs of hyperedges for n ≥ 3 and sufficiently large d . Using Gale transform and Ham-Sandwich theorem, we improve this lower bound to ( 2^d) ( n/2) ^d( (n-1)/2 ) ^d for n ≥ 3 and sufficiently large d .