Estimating the chromatic numbers of Euclidean space by convex minimization methods

The chromatic numbers of the Euclidean space with forbidden distances are investigated (that is, the minimum numbers of colours necessary to colour all points in so that no two points of the same colour lie at aforbidden distance from each other). Estimates for the growth exponents of the chromatic numbers as are obtained. The so-called linear algebra method which has been developed is used for this. It reduces the problem of estimating the chromatic numbers to an extremal problem. To solve this latter problem afundamentally new approach is used, which is based on the theory of convex extremal problems and convex analysis. This allows the required estimates to be found for any. For these estimates are found explicitly; they are the best possible ones in the framework of the method mentioned above. © 2009 Russian Academy of Sciences, (DoM) and London Mathematical Society, Turpion Ltd.