New examples of c0-saturated Banach spaces II

For every Banach space Z with a shrinking unconditional basis satisfying an upper p-estimate for some p>1, an isomorphically polyhedral Banach space is constructed which has an unconditional basis and admits a quotient isomorphic to Z. It follows that reflexive Banach spaces with an unconditional basis and non-trivial type, Tsirelson's original space and (∑c0)ℓp for p∈(1,∞), are isomorphic to quotients of isomorphically polyhedral Banach spaces with unconditional bases.