Estimating the chromatic numbers of Euclidean space by convex minimization methods
- M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow (Russian Federation)
The chromatic numbers of the Euclidean space R{sup n} with k forbidden distances are investigated (that is, the minimum numbers of colours necessary to colour all points in R{sup n} 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{yields}{infinity} 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.
- OSTI ID:
- 21301596
- Journal Information:
- Sbornik. Mathematics, Vol. 200, Issue 6; Other Information: DOI: 10.1070/SM2009v200n06ABEH004019; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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