On the colouring of spheres embedded in $ \mathbb R^n$
Андрей Борисович Купавский · Sbornik Mathematics · 2011
The work concerns the well-known problem of identifying the chromatic number of the space?, that is, finding the minimal number of colours required to colour all points of the space in such a?way that any two points at distance one from each other have different colours. A?new quantity generalising the chromatic number is introduced in the paper, namely, the speckledness of a?subset in a?fixed metric space. A?series of lower bounds for the speckledness of spheres is derived. These bounds are used to obtain general results lifting lower bounds for the chromatic number of a?space to higher dimensions, generalising the well-known 'Moser-Raisky spindle'. As a?corollary of these results, the best known lower bound for the chromatic number is obtained, and furthermore, the known bound is reproved in several different ways. Bibliography: 23 titles.