Borsuk number in four dimensions is at most seven
Abstract
We prove that every bounded set of diameter one in four-dimensional Euclidean space can be covered by seven balls of diameter strictly smaller than $0.9999$. In particular, the Borsuk number satisfies b4≤7. The proof combines geometric arguments based on contact points of the circumball with an exact finite computation. For small and large circumradius, the covering centres are constructed analytically. In the remaining range, a suitable triangle of contact points reduces the problem to a four-parameter family of ball polytopes. The parameter cube is subdivided into 23494 boxes. For each parameter box, candidate centres are found by nonlinear optimization, and the resulting covering is verified using exact rational interval arithmetic and paraboloid lifting.
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BibTeX
@misc{arman2026borsuk,
title = {{Borsuk number in four dimensions is at most seven}},
author = {Andrii Arman and Andriy Bondarenko and Andriy Prymak and Danylo Radchenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39078},
eprint = {2610.39078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39078},
}
RIS
TY - UNPB
TI - Borsuk number in four dimensions is at most seven
AU - Andrii Arman
AU - Andriy Bondarenko
AU - Andriy Prymak
AU - Danylo Radchenko
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39078
UR - https://arxiv.org/abs/2610.39078
AB - We prove that every bounded set of diameter one in four-dimensional Euclidean space can be covered by seven balls of diameter strictly smaller than $0.9999$. In particular, the Borsuk number satisfies b_4≤7. The proof combines geometric arguments based on contact points of the circumball with an exact finite computation. For small and large circumradius, the covering centres are constructed analytically. In the remaining range, a suitable triangle of contact points reduces the problem to a four-parameter family of ball polytopes. The parameter cube is subdivided into 23494 boxes. For each parameter box, candidate centres are found by nonlinear optimization, and the resulting covering is verified using exact rational interval arithmetic and paraboloid lifting.
ER -