On Lower Bounds for the Chromatic Number of Spheres
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Estimates of the chromatic numbers of spheres are studied. The optimality of the choice of the parameters of the linear-algebraic method used to obtain these estimates is investigated. For the case of (0, 1)-vectors, it is shown that the parameters chosen in previous results yield the best estimate. For the case of (−1, 0, 1)-vectors, the optimal values of the parameters are obtained; this leads to a significant refinement of the estimates of the chromatic numbers of spheres obtained earlier.
Keywordschromatic number of spheres linear-algebraic method Frankl–Wilson theorem Nelson–Hadwiger problem distance graphs
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- 3.A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters,” in Discrete Geometry and Algebraic Combinatorics, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2014), Vol. 625, pp. 93–109.Google Scholar
- 4.A. M. Raigorodskii, “Coloring Distance Graphs and Graphs of Diameters,” in Thirty Essays on Geometric Graph Theory (Springer, New York, 2013), pp. 429–460.Google Scholar
- 14.P. Erdős and R. L. Graham, Problem Proposed at the 6th Hungarian Combinatorial Conference (Eger, 1981).Google Scholar
- 21.A. M. Raigorodskii, Linear–Algebraic Method in Combinatorics (MTsNMO,Moscow, 2015) [in Russian].Google Scholar
- 26.E. I. Ponomarenko and A. M. Raigorodskii, “New upper bounds for the independence numbers of graphs with vertices in −1, 0, 1n and their applications to problems of the chromatic numbers of distance graphs,” Mat. Zametki 96 (1), 138–147 (2014) [Math. Notes 96 (1), 140–148 (2014)].CrossRefGoogle Scholar