Publication

Abstract : In this paper, we study the d-dimensional rectilinear drawings of the complete d-uniform hypergraph . Anshu et al. (2017) [3] used Gale transform and Ham-Sandwich theorem to prove that there exist crossing pairs of hyperedges in such a drawing of . We improve this lower bound by showing that there exist crossing pairs of hyperedges in a d-dimensional rectilinear drawing of . We also prove the following results. 1. There are crossing pairs of hyperedges in a d-dimensional rectilinear drawing of when its 2d vertices are either not in convex position in or form the vertices of a d-dimensional convex polytope that is t-neighborly but not -neighborly for some constant independent of d. 2. There are crossing pairs of hyperedges in a d-dimensional rectilinear drawing of when its 2d vertices form the vertices of a d-dimensional convex polytope that is -neighborly for some constant independent of d.