## Abstract

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 ℝ^{n} 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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## 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 ℝ^{n}
.
*Combinatorica* **32, **111–123 (2012). https://doi.org/10.1007/s00493-012-2709-9

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### Mathematics Subject Classification (2000)

- 52C10
- 05C15