Banach Spaces of Bounded Szlenk Index II
  For every $\alpha<\omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $\omega^{\alpha\omega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $\omega^{\alpha\omega}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates. 
D. Freeman, E. Odell, Th. Schlumprecht, and A. Zsak