Polytopes with large transversal ratio

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.

Figures (1)

• Figure 1: A drawing of the vertices of $\mathop{\mathrm{HJ}}\nolimits(3,2)$ and the curve $Z_L^\epsilon$ for each combinatorial line $L$ of $\mathop{\mathrm{HJ}}\nolimits(3,2)$. The blue dots are points of $P$ and the red dots are points of $P^\epsilon$.

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Conjecture 1.4
• Remark 1.5
• Corollary 1.7
• Corollary 1.10
• Theorem 2.1: Density Hales-Jewett Theorem density-Hales-Jewett_original
• Theorem 2.2
• Lemma 2.3
• proof