Modified mixed Tsirelson spaces

We study the modified and boundedly modified mixed Tsirelson spaces $T_M[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ and $T_{M(s)}[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ respectively, defined by a subsequence $({\cal F}_{k_n})$ of the sequence of Schreier families $({\cal F}_n)$. These are reflexive asymptotic $\ell_1$ spaces with an unconditio- nal basis $(e_i)_i$ having the property that every sequence $\{ x_i\}_{i=1}^n$ of normalized disjointly supported vectors contained in $\langle e_i\rangle_{i=n}^{\infty }$ is equivalent to the basis of $\ell_1^n$. We show that if $\lim\theta_n^{1/n}=1$ then the space $T[({\cal F}_n,\theta_n) _{n=1}^{\infty }]$ and its modified variations are totally incomparable by proving that $c_0$ is finitely disjointly representable in every block subspace of $T[({\cal F}_n, \theta_n)_{n=1}^{\infty }]$. Next, we present an example of a boundedly modified mixed Tsirelson space $X_{M(1),u}$ which is arbitrarily distortable. Finally, we construct a variation of the space $X_{M(1),u}$ which is hereditarily indecomposable.