Lower Bounds for Measurable Chromatic Numbers

Bachoc, Christine; Nebe, Gabriele; Filho, Fernando Mário de Oliveira; Vallentin, Frank

doi:10.1007/s00039-009-0013-7

Cited by 36 publications

(62 citation statements)

References 23 publications

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“…It is interesting to notice that this lower bound coincides with the one provided by Bachoc et al [2,Corollary 8.2], albeit with a shift of one dimension. This shift is due to the fact that Bachoc et al [2] study the problem of sets avoiding one distance on the (n − 1)-dimensional unit sphere S n−1 ⊆ R n , and the lower bound for the measurable chromatic number χ m (R n ) was obtained by upper bounding the density of sets in the unit sphere which avoid the distance d where d goes to zero. So, we see now that this limit process gives a lower bound for the measurable chromatic number of R n−1 and not only for R n .…”

Section: Sets Avoiding One Distancesupporting

confidence: 83%

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Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb{R}^n$

Filho¹,

Vallentin²

2010

J. Eur. Math. Soc.

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Abstract. 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 short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss, Bourgain and Falconer about sets avoiding many distances.

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Section: Sets Avoiding One Distancesupporting

confidence: 83%

“…, 24 based on a strengthening of our main theorem by extra inequalities. These new upper bounds for m 1 (R n ) imply by (2) new lower bounds for χ m (R n ) in dimensions 3, . .…”

Section: Introductionmentioning

confidence: 65%

Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb{R}^n$

Filho¹,

Vallentin²

2010

J. Eur. Math. Soc.

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“…Our first main result is that α(3) < 0.313. The proof involves tightening a Delsarte-type linear programming upper bound (see [2,5,7,8]) by adding combinatorial constraints.…”

Section: Clearly C M (N) ≥ C(n) Remarkably It Is Still Open If Thementioning

confidence: 99%

Spherical Sets Avoiding a Prescribed Set of Angles

DeCorte

Pikhurko

2015

Int Math Res Notices

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Let X be any subset of the interval [−1, 1]. A subset I of the unit sphere in R n will be called X-avoiding if u, v / ∈ X for any u, v ∈ I . The problem of determining the maximum surface measure of a {0}-avoiding set was first stated in a 1974 note by Witsenhausen;there the upper bound of 1/n times the surface measure of the sphere is derived from a simple averaging argument. A consequence of the Frankl-Wilson theorem is that this fraction decreases exponentially, but until now the 1/3 upper bound for the case n= 3 has not moved. We improve this bound to 0.313 using an approach inspired by Delsarte's linear programming bounds for codes, combined with some combinatorial reasoning. In the second part of the paper, we use harmonic analysis to show that, for n≥ 3, there always exists an X-avoiding set of maximum measure. We also show with an example that a maximizer need not exist when n= 2.

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“…Again, we aim at lower bounding the measurable chromatic number χ m (G (S n−1 , D)). Here we extend the technique of Bachoc, Nebe, Oliveira and Vallentin [6], who gave a formulation for the ϑ-number of G(S n−1 , D). However, they showed how to compute it only in the case that D is finite.…”

Section: Graphs On the Unit Spherementioning

confidence: 94%

Spectral bounds for the independence ratio and the chromatic number of an operator

et al. 2014

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Abstract. We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L 2 -space 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.

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Lower Bounds for Measurable Chromatic Numbers

Cited by 36 publications

References 23 publications

Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb{R}^n$

Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb{R}^n$

Spherical Sets Avoiding a Prescribed Set of Angles

Spectral bounds for the independence ratio and the chromatic number of an operator

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