in [@BKL] reflexivity is not needed here.

Assume that $X$ is a separable Banach space and that $\max\big({\text{\rm Sz}}(X),{\text{\rm Sz}}(X^*)\big)> \omega^\alpha$, for some countable ordinal $\alpha$. Then $({\mathcal S}_\alpha,d_{1,\alpha})$ embeds bi-Lipschitzly into $X$ and $X^*$.

We will deduce one direction of Theorem A from James’s characterization of reflexive Banach spaces [@Ja0], and show that for any non reflexive Banach space $X$ and any $\alpha<\omega_1$ there is a map $\Phi_\alpha:{\mathcal S}_\alpha \to X$ which satisfies . The converse will follow from the following result.

Assume that $X$ is a reflexive and separable Banach space. Let $  \xi<\omega_1$ and put $\beta=\omega^{\omega^\xi}$. If for some numbers $C>c>0$ there exists a $c$-lower-$d_{\alpha,\infty}$ and $C$-upper-$d_{\alpha,1}$ embedding of ${\mathcal S}_{\beta^2}$ into $X$, then ${\text{\rm Sz}}(X)> \omega^{\beta}$ or ${\text{\rm Sz}}(X^*)>\beta$.

Theorem C, and thus the missing part of Theorem A, will be shown in Section \[S:12\], Theorem \[T:12.6\]. Combining Theorems B and C, we obtain the following characterization of certain bounds of the Szlenk index of $X$ and its dual $X^*$. This result extends the main result of [@BKL] to separable and reflexive Banach spaces with higher order Szlenk indices.

\[C:1.1\] Assume that $\omega<\alpha<\omega_1$ is an ordinal for which $\omega^\alpha=\alpha$. Then the following statements are equivalent for 