Lower Bounds for Measurable Chromatic Numbers

By C. Bachoc, Gabriele Nebe, Fernando Mário Oliveira Filho, Frank Vallentin. 2008

53 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/2ffb71ffd6eb51a2697446e00e95248fe255b8d9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Christopher J. Hillar
2. Lek-Heng Lim
3. Boris N. Khoromskij
4. Liqun Qi
5. André Uschmajew
6. Ivan V. Oseledets
7. Anima Anandkumar
8. Yaroslav N. Shitov

In Discrete Mathematics and Combinatorics

1. Lek-Heng Lim

In Numerical Analysis

1. Venera Khoromskaia
2. Charalampos E. Tsourakakis

Citations in the corpus

• Most Tensor Problems Are NP-Hard (2009, background): “shown, assuming the Unique Games Conjecture, that Goemans-Williamson’s approximation factor is best possible [Khot et al. 2007]. We refer the reader to [Alon and Naor 2004; Bachoc and Vallentin 2008; Bachoc et al. 2009; Briet et al. 2010a; Briet et al. 2010b; He et al. 2010] for some recent work in ”