Antipodal paths in covers of spheres

Abstract

In this note we show that if the sphere ${\mathbb{S}}^n$ is covered by $k$ open sets with $n\ge 2k-2$, 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.

arXiv comment3 pages. Comments (by email) welcome