Distorting mixed Tsirelson spaces

Any regular mixed Tsirelson space T(theta(n),S-n)(N) for which theta(n)/the ta(n) --> 0, where theta = lim(n) theta(n)(1/n), is shown to be arbitrarily distortable. Certain asymptotic l(1) constants for those and other mixed T sirelson spaces are calculated. Also, a combinatorial result on the Schreie r 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-sp ectrums Delta(X) and Delta(X, (e(i))) coincide.