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arXiv · 2607.13964

Antipodal paths in covers of spheres

David Ellis·Maria-Romina Ivan·Imre Leader·John M. Mackay
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Abstract

In this note we show that if the sphere $\mathbb{S}^n$ is covered by $k$ open sets with $n \geq 2k-2$, then one of these sets contains a path with antipodal endpoints. This is best possible in the sense that the statement fails for $n < 2k-2$. The result can be seen as a spherical analogue of a well-known conjecture of Norine on edge-colourings of the discrete hypercube.

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David Ellis, Maria-Romina Ivan, Imre Leader, John M. Mackay. 2026-07-15. Antipodal paths in covers of spheres. https://arxiv.org/abs/2607.13964

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