Unbounded weak chromatic number for geometric tetrahedral complexes
Let $H = (V, E)$ be a finite $4$-uniform hypergraph. A geometric embedding of $H$ into $\mathbb{R}^3$ is an injection $\phi : V \to \mathbb{R}^3$ such that each $\operatorname{conv}\phi(e)$ is a nondegenerate tetrahedron and $$\operatorname{conv}\phi(e) \cap \operatorname{conv}\phi(f) = \operatorname{conv}\phi(e \cap f) \qquad (e, f \in E),$$ with $\operatorname{conv}\varnothing = \varnothing$. The weak chromatic number of $H$ is the least number of colors of $V$ for which no hyperedge is monochromatic. Heise, Panagiotou, Pikhurko and Taraz proved that the weak chromatic number of $3$-uniform hypergraphs geometrically embeddable in $\mathbb{R}^3$ is unbounded. Lee and Nevo constructed $4$-uniform hypergraphs geometrically embeddable in $\mathbb{R}^3$ with weak chromatic number at least $3$, and proved that it is unbounded when piecewise-linear embeddings are allowed. Dobbins and Lee proved that for every $d \ge 4$ the weak chromatic number of $(d+1)$-uniform hypergraphs geometrically embeddable in $\mathbb{R}^d$ is unbounded.
Problem. Is the weak chromatic number unbounded over $4$-uniform hypergraphs admitting such an embedding into $\mathbb{R}^3$, that is, for every integer $m$, is there one with weak chromatic number greater than $m$?
Source: Heise, Panagiotou, Pikhurko and Taraz (2014), §4, who note that for (d+1)-uniform hypergraphs embeddable in ℝ^d a finite bound is still possible; asked by Lee and Nevo (2026), Problem 1.2(1), for every d ≥ 3. Stated for d = 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, geometric hypergraphs, simplicial complexes, weak colorings
References
@article{heise2014,
author = {Heise, Carl Georg and Panagiotou, Konstantinos and Pikhurko, Oleg and Taraz, Anusch},
title = {Coloring $d$-embeddable $k$-uniform hypergraphs},
journal = {Discrete Comput. Geom.},
volume = {52},
number = {4},
year = {2014},
pages = {663--679},
doi = {10.1007/s00454-014-9641-2},
eprint = {1209.4879},
archivePrefix = {arXiv},
}
@article{lee2026,
author = {Lee, Seunghun and Nevo, Eran},
title = {On colorings of hypergraphs embeddable in $\mathbb{R}^d$},
journal = {Discrete Comput. Geom.},
volume = {76},
number = {3},
year = {2026},
pages = {1942--1959},
doi = {10.1007/s00454-026-00856-4},
eprint = {2307.14195},
archivePrefix = {arXiv},
}
@misc{dobbins2026,
author = {Dobbins, Michael Gene and Lee, Seunghun},
title = {Polytopes with large transversal ratio},
year = {2026},
eprint = {2603.16298},
archivePrefix = {arXiv},
}
@misc{aim-tetrahedral-weak-chromatic,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Unbounded weak chromatic number for geometric tetrahedral complexes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/613-tetrahedral-weak-chromatic.md},
note = {Entry 613 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00908, "Unbounded weak chromatic number for geometric tetrahedral complexes", clopen.solutions.