$k$-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. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich theorem to prove that there exist $\Omega \left(2^d\right)$ crossing pairs of hyperedges in such a drawing of $K_{2d}^d$. We improve this lower bound by showing that there exist $\Omega \left(2^d \sqrt{ d}\right)$ crossing pairs of hyperedges in a $d$-dimensional rectilinear drawing of $K_{2d}^d$. We also prove the following results. 1. There are $\Omega \left(2^d {d^{3/2}}\right)$ 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 $\mathbb{R}^d$ or form the vertices of a $d$-dimensional convex polytope that is $t$-neighborly but not $(t+1)$-neighborly for some constant $t\geq1$ independent of $d$. 2. There are $\Omega \left(2^d {d^{5/2}}\right)$ 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 $(\lfloor{d/2}\rfloor-t')$-neighborly for some constant $t' \geq 0$ independent of $d$.