Polytopes with large transversal ratio

TL;DR

This paper constructs an infinite family of high-dimensional polytopes with a transversal ratio approaching 1, demonstrating that their weak chromatic number is unbounded for dimensions five and higher.

Contribution

It introduces a new family of polytopes with large transversal ratios, improving previous lower bounds and showing unbounded weak chromatic numbers in higher dimensions.

Findings

Transversal ratio approaches 1 for the constructed polytopes.

Weak chromatic number is unbounded for all dimensions ≥ 5.

Improves previous lower bounds on transversal ratios.

Abstract

The transversal ratio of a polytope $P$ is the minimum proportion of vertices of $P$ required to intersect each facet of $P$. The weak chromatic number of $P$ is the minimum number of colors required to color the vertices of $P$ so that no facet is monochromatic. We will construct an infinite family of $d$-polytopes for each $d\geq 5$ whose transversal ratio approaches 1 as the number of vertices grows. In particular, this implies that the weak chromatic number for $d$-polytopes is unbounded for each $d\geq 5$. The previous best known lower bounds on the supremum of the transversal ratio for $d$-polytopes for $d\geq 5$ were 2/5 for odd $d$ by Novik and Zheng, and 1/2 for even $d$ by Holmsen, Pach, and Tverberg. In the case of simplicial $(d-1)$-spheres, the best known lower bounds were 1/2 for $d=5$ and $6/11$ for $d=6$ by Novik and Zheng.