AI 中文总结
研究当球面\(\mathbb{S}^n\)被\(k\)个开集覆盖时的情况，在\(n\geq2k - 2\)条件下，证明其中一个开集含对映端点路径，此为诺林猜想的球面类似物，且给出了\(n\)的临界条件。
AI 中文摘要
在本笔记中，我们证明如果\(n\geq2k - 2\)时，球面\(\mathbb{S}^n\)被\(k\)个开集覆盖，那么其中一个集合包含一条具有对映端点的路径。当\(n < 2k - 2\)时该陈述不成立，此结果是离散超立方体边着色的著名诺林猜想的球面类似物。
英文摘要
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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