Estimating the chromatic numbers of Euclidean space by convex minimization methods

Gorskaya, Elena Sergeevna; Митричева, Ирина Михайловна; Protasov, Vladimir Yu.; Райгородский, А. М.

doi:10.1070/sm2009v200n06abeh004019

Cited by 24 publications

(21 citation statements)

References 16 publications

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“…Gorskaya, Mitricheva, Protasov, and Raigorodskii [7] described a method for determining the optimal number ζ k for which there is a sequence of graphs {G n } ∞ n=1 satisfying the conditions…”

Section: Application Of the Methodsmentioning

confidence: 99%

“…The results which can be obtained for k = 2, 3 are presented in the table (the optimal values for ζ k are taken from the paper [7]). …”

Section: Application Of the Methodsmentioning

confidence: 99%

“…In 2009, Gorskaya, Mitricheva, Protasov, and Raigorodskii [7] proposed a method for determining asymptotic lower bounds for the quantity χ(R n ; k) of the form…”

Section: Introductionmentioning

confidence: 99%

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On the chromatic number of Euclidean space with two forbidden distances

Berdnikov

Райгородский

2014

Math Notes

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The work is devoted to to one of the variations of the Hadwiger-Nelson problem on the chromatic number of the plane. In this formulation one needs to find for arbitrarily small ε the least possible number of colors needed to color the Euclidean plane in such a way that any two points, the distance between which belongs to the interval [1 − ε, 1 + ε], are colored differently. The hypothesis proposed by G. Exoo in 2004, states that for arbitrary positive ε at least 7 colors are required. Also, with a sufficiently small ε the number of colors is exactly 7. The main result of this paper is the proof of this hypothesis.

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Section: Application Of the Methodsmentioning

confidence: 99%

“…The results which can be obtained for k = 2, 3 are presented in the table (the optimal values for ζ k are taken from the paper [7]). …”

Section: Application Of the Methodsmentioning

confidence: 99%

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On the chromatic number of Euclidean space with two forbidden distances

Berdnikov

Райгородский

2014

Math Notes

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“…For a comprehensive survey of these (and many other) discrete geometry problems see [3]; a survey of results concerning unit-distance graphs can be found in [4]- [6]. Some isolated properties of A-distance graphs were studied for finite sets A (see [4], [7], [8]). 1 In literature unit-distance graphs and objects of similar nature appear under different names, such as linkages ( [9]), embeddable ( [10]), or realizable ( [9]) graphs.…”

Section: Problem Statement and Motivationmentioning

confidence: 99%

On complexity of multidistance graph recognition in R1

Tikhomirov

2017

Electronic Notes in Discrete Mathematics

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“…[3]- [10]); изучались хроматические чис-ла пространств с несколькими (и даже c бесконечно многими) запрещенными расстояниями (см. [7], [8], [11]- [14]). А одним из наиболее интересных направле-ний исследований стала так называемая евклидова теория Рамсея.…”

Section: о хроматическом числе пространства с запрещенным равносторонunclassified