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 $1$?
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.
https://en.wikipedia.org/wiki/Gleason%27s_theorem
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