Abstract
The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces.
In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10, . . . , 24 and we give a new proof that it grows exponentially with the dimension.
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Acknowledgments
We thank Dion Gijswijt, Gil Kalai, Tom Koornwinder, Pablo Parrilo, and Lex Schrijver for their helpful comments.
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The third author was partially supported by CAPES/Brazil under grant BEX 2421/04-6. The fourth author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant SCHU 1503/4.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bachoc, C., Nebe, G., de Oliveira Filho, F.M. et al. Lower Bounds for Measurable Chromatic Numbers. Geom. Funct. Anal. 19, 645–661 (2009). https://doi.org/10.1007/s00039-009-0013-7
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DOI: https://doi.org/10.1007/s00039-009-0013-7
Keywords and phrases
- Nelson–Hadwiger problem
- measurable chromatic number
- semidefinite programming
- orthogonal polynomials
- spherical codes