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Suppose that $S^2$ is the unit sphere in $\mathbb{R}^3$.

Is there a function $f \colon S^2 \to \{0,1\}$ so that, for any orthonormal basis $(u,v,z)$, exactly one of the values $f(u)$, $f(v)$, and $f(z)$ equals $1$?

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    $\begingroup$ This looks very much like a Hadwiger-Nelson type problem for the unit sphere (at least in the following sense: if the unit sphere is $3$-colorable with forbidden distance equal to that from the pole to the equator, then your question has a positive answer). Such problems are notoriously difficult. But one thing that can easily be said is that there is a compactness phenomenon: if the answer is “no”, then there is a finite witness to this fact. $\endgroup$
    – Gro-Tsen
    Aug 22, 2023 at 20:15
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    $\begingroup$ It is known that the unit-sphere with this forbidden distance is 4-chromatic; see Simmons: The chromatic number of the sphere, 1976, or for a more recent, independent proof, arxiv.org/abs/1201.0486. Some newer results are in arxiv.org/abs/2203.08666. $\endgroup$
    – domotorp
    Aug 22, 2023 at 21:32
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    $\begingroup$ I think that this question is more similar to the finite version of Steinhaus's problem, see this manuscript: arxiv.org/abs/math/0603235. $\endgroup$
    – domotorp
    Aug 22, 2023 at 21:52
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    $\begingroup$ Compare Gleason's theorem https://en.wikipedia.org/wiki/Gleason%27s_theorem $\endgroup$ Aug 22, 2023 at 23:51
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    $\begingroup$ You should inform that this question is crossposted to avoid duplication of effort by people answering on different sites. $\endgroup$ Aug 23, 2023 at 10:16

1 Answer 1

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The answer is no. This was proven by Kochen and Specker in 1967 in the context of quantum mechanics, refuting a notion of hidden variables. They found 117 points on the sphere that cannot be assigned values 0 and 1 in the requested way. (This is a finite witness that Gro-Tsen alluded to. And following up on Gerald Edgar's comment, this discrete collection demonstrates a corollary of Gleason's continuous result.)

For a nice explanation and bibliographic details, see this Stanford Encyclopedia of Philosophy entry which also mentions subsequent results requiring fewer points and simpler $\mathbb{R}^4$ examples.

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    $\begingroup$ And for a direct reference to an explicit construction using only 33 points, Citeseer has a PDF of Peres, A. (1991). Two simple proofs of the Kochen-Specker theorem. Journal of Physics A: Mathematical and General, 24(4), L175. $\endgroup$ Aug 23, 2023 at 13:56
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    $\begingroup$ Unbelievable. This means that the chromatic number of the sphere with this radius was proved to be four a decade before Simmons's paper in quantum mechanical terminology. $\endgroup$
    – domotorp
    Aug 23, 2023 at 15:28

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