Estimation of the Szlenk index of reflexive Banach spaces using generalized Baernstein spaces

For each ordinal $\alpha <{\omega }_1$, we prove the existence of a separable, reflexive Banach space with a basis and Szlenk index ${\omega }^{\alpha +1}$ which is universal for the class of separable, reflexive Banach spaces $X$ such that the Szlenk indices $Sz(X),Sz(X^{\ast })$ do not exceed ${\omega }^{\alpha }$.