  For each ordinal $\alpha<\omega_1$, we prove the existence of a space with a
basis and Szlenk index $\omega^{\alpha+1}$ which is universal for the class of
spaces with Szlenk index not exceeding $\omega^\alpha$. Our proof involves
developing a characterization of which Banach spaces embed into spaces with an
FDD with upper Schreier space estimates.
