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Christine Bachoc(bachocmath.ubordeaux1.fr) Abstract: The Lov\'asz 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, twovariable 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. Keywords: NelsonHadwiger problem, measurable chromatic number, semidefinite programming, orthogonal polynomials, spherical codes Category 1: Linear, Cone and Semidefinite Programming (Semidefinite Programming ) Citation: arXiv:0801.1059v1 [math.CO] 7 Jan 2008 Download: [PDF] Entry Submitted: 01/15/2008 Modify/Update this entry  
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