Unextendible Product Bases

Let C denote the complex field. A vector v in the tensor product ?mi=1Cki is called a pure product vector if it is a vector of the form v1?v2??vm, with vi?Cki. A set F of pure product vectors is called an unextendible product basis if F consists of orthogonal nonzero vectors, and there is no nonzero pure product vector in ?mi=1Cki which is orthogonal to all members of F. The construction of such sets of small cardinality is motivated by a problem in quantum information theory. Here it is shown that the minimum possible cardinality of such a set F is precisely 1+?mi=1(ki?1) for every sequence of integers k1, k2, ?, km?2 unless either (i) m=2 and 2?{k1, k2} or (ii) 1+?mi=1(ki?1) is odd and at least one ki is even. In each of these two cases, the minimum cardinality of the corresponding F is strictly bigger than 1+?mi=1(ki?1).