  For each ordinal $\alpha< \omega_1$, we prove the existence of a separable,
reflexive Banach space with a basis and Szlenk index $\omega^{\alpha+1}$ which
is universal for the class of separable, reflexive Banach spaces $X$ such that
the Szlenk indices $Sz(X), Sz(X^*)$ do not exceed $\omega^\alpha$.
