Note on decomposition of spheres in Hilbert spaces (Q1082349)

Let \(\lambda\) be an infinite cardinal and \(S^{\lambda}\) and \(B^{\lambda}\) be the unit sphere and the unit ball of the Hilbert space of \(\lambda\) dimensions. For \(c\in (0,2)\) define the graphs \(S^{\lambda}_{\geq c}\) and \(B^{\lambda}_{\geq c}\) \((S^{\lambda}_{=c}\) and \(B^{\lambda}_{=c}\) respectively) with the vertex set \(S^{\lambda}\) and \(B^{\lambda}\) and two points x, y joined by an edge if this distance is at least c (equal to c respectively). In the present paper we study the chromatic number \(\chi\) of graphs \(S^{\lambda}_{\geq c}\), \(B^{\lambda}_{\geq c}\), \(S^{\lambda}_{=c}\), \(B^{\lambda}_{=c}\). We will show that \[ \chi (B^{\lambda}_{\geq c})=\chi (S^{\lambda}_{\geq c})=\chi (B^{\lambda}_{=c})=\chi (S^{\lambda}_{=c})=\min \{;\exp \exp...\exp m_{(k-times)}\geq \lambda \} \] for \(c\in (\sigma_ k,\sigma_{k+1})\) where \(\sigma_ 0=0\) and \(\sigma_ j=\sqrt{2-2 \cos (\frac{j}{j+1}\pi)}.\) The situation gets more complicated if we consider the graphs with edges of distance bigger than some \(c\in (0,2)\). For \(c\neq \sigma_ k\), \(k=0,1,..\). the chromatic number can be easily computed using above result but for \(c=\sigma_ k\), \(k=0,1,..\). our method cannot be used. Answering a question of P. Erdős we give here a partial answer and show that the unit sphere \(S^{\aleph_ 1}\) cannot be written as a union of countably many sets of diameters less or equal than \(\sqrt{2}\).