Lower bounds for measurable chromatic numbers

TL;DR

This paper extends the Lovasz theta function to provide lower bounds for the measurable chromatic number of distance graphs on compact metric spaces, including spheres, and derives new bounds for Euclidean spaces in high dimensions.

Contribution

It generalizes the Lovasz theta function to measurable chromatic numbers and solves an extremal problem involving Jacobi polynomials for spheres.

Findings

Derived new lower bounds for the measurable chromatic number in dimensions 10-24.

Proved exponential growth of the measurable chromatic number with dimension.

Solved an extremal problem related to Jacobi polynomials explicitly.

Abstract

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.