Some remarks on spreading models and mixed Tsirelson spaces

We prove that if a Banach space with a bimonotone shrinking basis does not contain $\ell_{1}^{\omega}$ spreading models but every block sequence of the basis contains a further block sequence which is a $c-\ell_{1}^{n}$ spreading model for every $n\in\mathbb{N}$, then every subspace has a further subspace which is arbitrarily distortable. We also prove that a mixed Tsirelson space $T[(\mathcal{S}_{n},\theta_{n})_{n}]$, such that $\theta_{n}\searrow 0$, does not contain $\ell_{1}^{\omega2}$ spreading models.