On the chromatic numbers of spheres in ℝn


Let χ(S n−1 r )) be the minimum number of colours needed to colour the points of a sphere S n−1 r of radius \(r \geqslant \tfrac{1} {2}\) in ℝn so that any two points at the distance 1 apart receive different colours. In 1981 P. Erdős conjectured that χ(S n−1 r )→∞ for all \(r \geqslant \tfrac{1} {2}\). This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S n−1 r ) ≥ n. In the same paper, Lovász claimed that if \(r < \sqrt {\frac{n} {{2n + 2}}}\), then χ(S n−1 r ) ≤ n+1, and he conjectured that χ(S n−1 r ) grows exponentially, provided \(r \geqslant \sqrt {\frac{n} {{2n + 2}}}\). In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S n−1 r ) grows exponentially for any \(r > \tfrac{1} {2}\).

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Correspondence to A. M. Raigorodskii.

Additional information

This work is done under the financial support of the following grants: the grant 09-01-00294 of Russian Foundation for Basic Research, the grant MD-8390.2010.1 of the Russian President, the grant NSh-8784.2010.1 supporting Leading scientific schools of Russia, a grant of Dynastia foundation.

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Raigorodskii, A.M. On the chromatic numbers of spheres in ℝn . Combinatorica 32, 111–123 (2012). https://doi.org/10.1007/s00493-012-2709-9

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Mathematics Subject Classification (2000)

  • 52C10
  • 05C15