Distorting Mixed Tsirelson Spaces

Any regular mixed Tsirelson space $T(\theta_n,S_n)_{\N}$ for which $\frac{\theta_n}{\theta^n} \to 0$, where $\theta=\lim_n \theta_n^{1/n}$, is shown to be arbitrarily distortable. Certain asymptotic $\ell_1$ constants for those and other mixed Tsirelson spaces are calculated. Also a combinatorial result on the Schreier families $(S_{\alpha})_{\alpha < \omega_1}$ is proved and an application is given to show that for every Banach space $X$ with a basis $(e_i)$, the two $\Delta$-spectrums $\Delta(X)$ and $\Delta(X,(e_i))$ coincide.