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.