Transversals and colorings of simplicial spheres

TL;DR

This paper investigates the transversal and chromatic numbers of simplicial spheres, providing new constructions with specific ratios and improving bounds on chromatic numbers, thus advancing understanding of geometric and combinatorial properties of simplicial complexes.

Contribution

It introduces two infinite constructions of simplicial spheres with specific transversal ratios and refines upper bounds on the chromatic number of facet hypergraphs for simplicial spheres.

Findings

Constructed infinitely many simplicial polytopes with transversal ratio exactly 2/(d+2)

Found simplicial 3-spheres with transversal ratio greater than 1/2

Improved upper bound on chromatic number of facet hypergraphs for d-dimensional spheres

Abstract

Motivated from the surrounding property of a point set in $\mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial $d$-spheres, we provide two infinite constructions. The first construction gives infintely many $(d+1)$-dimensional simplicial polytopes with the transversal ratio exactly $\frac{2}{d+2}$ for every $d\geq 2$. In the case of $d=2$, this meets the previously well-known upper bound $1/2$ tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than $1/2$. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for $d\geq 3$, the facet hypergraph $\mathcal{F}(\mathsf{K})$ of a $d$-dimensional simplicial sphere $\mathsf{K}$ has the chromatic number $\chi(\mathcal{F}(\mathsf{K})) \in O(n^{\frac{\lceil d/2\rceil-1}{d}})$, where $n$ is the number of vertices of $\mathsf{K}$. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz.