On lower bounds for the chromatic number of sphere
The last lower bound for the chromatic number of the unit distance graph, whose vertex set is a sphere of radius \(r\) in the \(n\)-dimensional Euclidean space is due to Raigorodskii, the second author of the paper under review. He showed that for any fixed \(r>\frac12\) the chromatic number grows exponentially with \(n\). His lower bound, however, did not depend on \(r\) for \(\frac{1}{\sqrt{2}}<r\). This paper provides improved lower bounds that do depend on \(r\) if \(\frac{1}{\sqrt{2}}<r<\frac{1}{\root{4}\of{2}}\).
scientific article; zbMATH DE number 6507789