  We present a method to obtain sets of vectors proving the Bell-Kochen-Specker
theorem in dimension $n$ from a similar set in dimension $d$ ($3\leq d<n\leq
2d$). As an application of the method we find the smallest proofs known in
dimension five (29 vectors), six (31) and seven (34), and different sets
matching the current record (36) in dimension eight.
