Distorting Mixed Tsirelson Spaces∗

Any regular mixed Tsirelson space T (θn, Sn)IN for which θn θn → 0, where θ = limn θ n , is shown to be arbitrarily distortable. Certain asymptotic `1 constants for those and other mixed Tsirelson spaces are calculated. Also a combinatorial result on the Schreier families (Sα)α<ω1 is proved and an application is given to show that for every Banach space X with a basis (ei), the two ∆-spectrums ∆(X) and ∆(X, (ei)) coincide.