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

TL;DR

This paper investigates the number of crossing pairs of hyperedges in a geometric embedding of complete balanced d-partite d-uniform hypergraphs, providing improved lower bounds using advanced geometric theorems.

Contribution

It introduces new lower bounds on hyperedge crossings in geometric hypergraph embeddings, utilizing the Generalized Colored Tverberg Theorem, Gale Transform, and Ham-Sandwich Theorem.

Findings

Established a lower bound of  ext{(}8/3 ext{)}^{d/2} ext{(}n/2 ext{)}^d((n-1)/2)^d for crossings.

Improved the lower bound to  ext{(}2^{d} ext{)}(n/2)^d((n-1)/2)^d using geometric theorems.

Demonstrated that the number of crossings grows exponentially with dimension d.

Abstract

In this paper, we study the embedding of a complete balanced $d$-partite $d$-uniform hypergraph with all its $nd$ vertices represented as points in general position in $\mathbb{R}^d$ and each hyperedge drawn as a 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 of the vertices 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 the Generalized Colored Tverberg Theorem, we observe that such an embedding of a complete balanced $d$-partite $d$-uniform hypergraph with $nd$ vertices contains $\Omega\left((8/3)^{d/2}\right){\left({n/2}\right)^d{\left((n-1)/2\right)}^d}$ crossing pairs of hyperedges for sufficiently large $n$ and $d$. Using the Gale Transform and the Ham-Sandwich Theorem, we improve this lower bound to $ \Omega\left(2^{d}\right){\left({n/2}\right)^d{\left((n-1)/2\right)}^d}$ for sufficiently large $n$ and $d$.