Bell-Kochen-Specker theorem: A proof with 18 vectors

Abstract We present a “state-independent” proof of the Bell-Kochen-Specker theorem using only 18 four-dimensional vectors, which is a record for this kind of proof. This set of vectors contains subses which allow us to develop a “state-specific” proof with ten vectors (also a record) and a “probabilistic” proof with seven vectors which reflects the algebraic structure of Hardy's nonlocality theorem.