Two-colouring the two-sphere

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 $$1$$?

• 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. Commented Aug 22, 2023 at 20:15
• 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. Commented Aug 22, 2023 at 21:32
• I think that this question is more similar to the finite version of Steinhaus's problem, see this manuscript: arxiv.org/abs/math/0603235. Commented Aug 22, 2023 at 21:52
• Compare Gleason's theorem https://en.wikipedia.org/wiki/Gleason%27s_theorem Commented Aug 22, 2023 at 23:51
• You should inform that this question is crossposted to avoid duplication of effort by people answering on different sites. Commented Aug 23, 2023 at 10:16

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.