  For each $n \in \mathbb{N}$ a Banach space $\mathfrak{X}_{0,1}^n$ is
constructed is having the property that every normalized weakly null sequence
generates either a $c_0$ or $\ell_1$ spreading models and every infinite
dimensional subspace has weakly null sequences generating both $c_0$ and
$\ell_1$ spreading models. The space $\mathfrak{X}_{0,1}^n$ is also
quasiminimal and for every infinite dimensional closed subspace $Y$ of
$\mathfrak{X}_{0,1}^n$, for every $S_1,S_2,\ldots,S_{n+1}$ strictly singular
operators on $Y$, the operator $S_1S_2\cdots S_{n+1}$ is compact. Moreover, for
every subspace $Y$ as above, there exist $S_1,S_2,\ldots,S_n$ strictly singular
operators on $Y$, such that the operator $S_1S_2\cdots S_n$ is non-compact.
