k-Sets and rectilinear crossings in complete uniform hypergraphs

In this paper, we study the d-dimensional rectilinear drawings of the complete d-uniform hypergraph K-2d(d). Anshu et al. (2017) [3] used Gale transform and Ham-Sandwich theorem to prove that there exist Omega(2(d)) crossing pairs of hyperedges in such a drawing of K-2d(d). We improve this lower bound by showing that there exist Omega(2(d)root d) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K-2d(d). We also prove the following results. 1. There are Omega(2(d)d(3/2)) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K-2d(d) when its 2d vertices are either not in convex position in R-d or form the vertices of a d-dimensional convex polytope that is t-neighborly but not (t + 1)-neighborly for some constant t >= 1 independent of d. 2. There are Omega(2(d)d(5/2)) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K-2d(d) when its 2d vertices form the vertices of a d-dimensional convex polytope that is (left perpendiculard/2right perpendicular - t')-neighborly for some constant t' >= 0 independent of d. (C) 2019 Elsevier B.V. All rights reserved.