Polytopes with large transversal ratio
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911522074132480 |
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| author | Dobbins, Michael Gene Lee, Seunghun |
| author_facet | Dobbins, Michael Gene Lee, Seunghun |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16298 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polytopes with large transversal ratio Dobbins, Michael Gene Lee, Seunghun Combinatorics Metric Geometry 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. |
| title | Polytopes with large transversal ratio |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2603.16298 |