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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), $Sz(X*)\le {\omega }^{\alpha }$.

For $1<p\le \infty$, we show the existence of a Banach space which is both injectively and surjectively universal for the class of all separable Banach spaces with an equivalent $p$-asymptotically uniformly smooth norm. We prove that this class is analytic complete in the class of separable Banach spaces. These results extend previous works by N. J. Kalton, D. Werner and O. Kurka in the case $p=\infty$.