Estimation of the Szlenk index of Banach Spaces via Schreier spaces

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.Comment: arXiv admin note: text overlap with arXiv:0809.3626 by other author