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LOWER BOUNDS FOR MEASURABLE CHROMATIC NUMBERS
, 2008
"... 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 ..."
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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, 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.
FOURIER ANALYSIS, LINEAR PROGRAMMING, AND DENSITIES OF DISTANCE AVOIDING SETS IN R n
, 808
"... Abstract. In this paper we derive new upper bounds for the densities of measurable sets in R n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the ..."
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Abstract. In this paper we derive new upper bounds for the densities of measurable sets in R n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2,..., 24. This gives new lower bounds for the measurable chromatic number in dimensions 3,...,24. We apply it to get a new, short proof of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances. 1.
SPECTRAL BOUNDS FOR THE INDEPENDENCE RATIO AND THE CHROMATIC NUMBER OF AN OPERATOR
"... Abstract. We define the independence ratio and the chromatic number for bounded, selfadjoint operators on an L2space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numer ..."
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Abstract. We define the independence ratio and the chromatic number for bounded, selfadjoint operators on an L2space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the unit sphere. 1.