Some remarks on spreading models and mixed Tsirelson spaces

Abstract

We prove that if a Banach space with a bimonotone shrinking basis does not contain l w 1 spreading models but every block sequence of the basis contains a further block sequence which is a c - l n 1 spreading model for every n E N, then every subspace has a further subspace which is arbitrarily distortable. We also prove that a mixed Tsirelson space T[(S n ,θ n ) n ], such that θ n ? 0, does not contain l w2 1 spreading models.