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 $omega^{alphaomega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $omega^{alphaomega}$. 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.