Sphere Chromatic Number

The sphere chromatic number of a hypersphere is the smallest number of colors needed to color its points so that points at Euclidean distance 1 receive different colors. For the hypersphere $S_R^n={x in R^(n+1):||x||=R}$ , it is denoted and is the chromatic number of the Euclidean distance graph on with forbidden distance 1 (Prosanov 2018).

The corresponding question for the Euclidean plane is the Hadwiger-Nelson problem (Prosanov 2018).

Bikeev et al. (2026) improved the asymptotic upper bound for every fixed . They proved that

as . The exponential base is smaller than the earlier bound of Prosanov (2018) throughout this range and tends to as .

Bikeev et al. (2026) state that GPT-6.1 Sol supplied the main steps of the proof, which the authors independently verified and simplified.

References

Bikeev, A. I.; Voronov, V. A.; and Raigorodskii, A. M. "An Improved Upper Bound on the Chromatic Number of the

-Dimensional Sphere." 23 Oct 2026. https://arxiv.org/abs/2610.32408.

Prosanov, R. "Chromatic Numbers of Spheres." Disc. Math. 341, 3123-3133, 2018. https://doi.org/10.1016/j.disc.2018.07.014.

Sphere Chromatic Number

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