Lovász' theorem on the chromatic number of spheres revisited (Q276666)

The chromatic number of spheres \(\chi(S_r^{n-1})\) was introduced by Erdős in 1981 to be the minimum number of colors required to color all points of an \((n-1)\)-sphere \(S_r^{n-1}\) of radius \(r\) in \(\mathbb{R}^n\) so that no two points of the same color are distance 1 apart in the ambient Euclidean space \(\mathbb{R}^n\). The following is proved in this note. Let \(\omega\) be any function of a positive integer argument \(n\) which tends to infinity as \(n \to \infty\), and let \(r\) be any function of \(n\) taking a value at least \(\frac12 + \frac{\omega}{n}\) at each \(n\). Then \(\chi(S_{r(n)}^{n-1}) \to \infty\) as \(n \to \infty\). This result is slightly weaker than Lovász' earlier result that \(\chi(S_r^{n-1}) \geq n\) for any \(r > \frac12\) and any \(n\). Yet, the present proof is much shorter.