Uniqueness of unconditional bases in c(0)-products

We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation ) then so does c(0)(X). We also give some positive results including a simp ler proof that c(0)(l(1)) has a unique unconditional basis a unique uncondi tional basis and a proof that c(0)(l(pn)(Nn)) has a unique unconditional ba sis when p(n) down arrow 1, Nn+1 greater than or equal to 2N(n) and (p(n) - p(n+l)) log N-n remains bounded.