`1-spreading Models in Subspaces of Mixed Tsirelson Spaces

We investigate the existence of higher order `-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space X = T [( n;Sn) 1 n=1]: (1) Every block subspace of X contains an `1-S!-spreading model, (2) The Bourgain `-index Ib(Y ) = I(Y ) > ! ! for any block subspace Y of X, (3) limm lim supn m+n= n > 0 and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions holds, then X is arbitrarily distortable.