On strongly asymptotic $\ell_p$ spaces and minimality
  We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace). In particular such spaces embed into l_p. 
S. J. Dilworth, V. Ferenczi, Denka Kutzarova and E. Odell