Counterexamples to Borsuk's conjecture on spheres of small radius (Q630237)

The author defines for finite sets \(\Omega \subset \mathbb{R}^n\) with diameter 1 the graph of diameters \(G_\Omega =(\Omega, E_\Omega)\) by \(E_\Omega = \{\{x,y\}: |x-y|= 1\}\). Thus, the Borsuk number of \(\Omega\) (the minimum number of parts with smaller diameter into which \(\Omega\) can be partitioned) equals the chromatic number \(\chi(G_\Omega)\) of the graph \(G_\Omega\). In the present short paper, the author proves the following Theorem: For any \(r>\sqrt{3/8}\), there exists an \(n_0\) such that, for every \(n \geq n_0\), a sphere \(S^{n-1}_r \subset \mathbb{R}^n\) of radius \(r\) contains a set \(\Omega\) for which \(\chi(G_\Omega)>n+1\). This theorem shows that counterexamples to Borsuk's conjecture can be constructed on spheres with radii substantially smaller than \(1/\sqrt2\). However, examples are missed.