The Lovasz 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, two-variable 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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Birkhäuser
Geometric and Functional Analysis
Networks and Optimization

Bachoc, C, Nebe, G, de Oliveira Filho, F.M, & Vallentin, F. (2009). Lower bounds for measurable chromatic numbers. Geometric and Functional Analysis, 19(3), 645–661.