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For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index ${\omega }^{\alpha +1}$ which is universal for the class of separable Banach 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.

For each ordinal α < ω₁, we prove the existence of a separable, reflexive Banach space W with a basis so that Sz(W), $Sz(W*)\le {\omega }^{\alpha +1}$ which is universal for the class of separable, reflexive Banach spaces X satisfying Sz(X), $Sz(X*)\le {\omega }^{\alpha }$.