Bell-Kochen-Specker theorem for any finite dimension n>=3

The Bell-Kochen-Specker theorem against non-contextual hidden variables can be proved by constructing a finite set of 'totally non-colourable' directions, as Kochen and specker did in a Hilbert space of dimension n = 3. We generalize Kochen and Specker's set to Hilbert spaces of any finite dimension n greater than or equal to 3, in a three-step process that shows the relationship between different kinds of proofs ('continuum', 'probabilistic', 'state-specific' and 'state-independent') of the Bell-Kochen-specker theorem. At the same time, this construction of a totally noncolourable set of directions in any dimension explicitly solves the question raised by Zimba and Penrose about the existence of such a set for n = 5.