A Lower Bound on the Crossing Number of Uniform Hypergraphs

Abstract

In this paper, we consider the embedding of a complete $d$-uniform geometric hypergraph with $n$ vertices in general position in ${\mathbb{R}}^d$, where each hyperedge is represented as a $(d-1)$-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contains a common point in the relative interior of the simplices corresponding to them. As a corollary of the Van Kampen-Flores Theorem, it can be seen that such a hypergraph contains $\Omega (\frac{2^d}{\sqrt{d}})$ $\left(\genfrac{}{}{0ex}{}{n}{2d}\right)$ crossing pairs of hyperedges. Using Gale Transform and Ham Sandwich Theorem, we improve this lower bound to $\Omega (\frac{2^d\log d}{\sqrt{d}})$ $\left(\genfrac{}{}{0ex}{}{n}{2d}\right)$.

Cited by in corpus (3)

• On the Rectilinear Crossing Number of Complete Uniform Hypergraphs
• $k$-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs
• Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs