Lower bounds for measurable chromatic numbers

Details Lower bounds for measurable chromatic numbers Lower bounds for measurable chromatic numbers

2008-01-07

arxiv.org/abs/0801.1059v3

Geom. Funct. Anal. 19 (2009), 645-661

Mathematics

Combinatorics

18 pages, (v3) Section 8 revised and some corrections, to appear in Geometric and Functional Analysis

Scientific paper

10.1007/s00039-009-0013-7

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 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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