Estimating the chromatic numbers of Euclidean space by convex minimization methods

The chromatic numbers of the Euclidean space Rn with k forbidden distances are investigated (that is, the minimum numbers of colours necessary to colour all points in Rn so that no two points of the same colour lie at a forbidden distance from each other). Estimates for the growth exponents of the chromatic numbers as n→∞ 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 a fundamentally 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 k. For k≤20 these estimates are found explicitly; they are the best possible ones in the framework of the method mentioned above. Bibliography: 18 titles.