Prescribed Szlenk index of separable Banch spaces

In a previous work, the first named author described the set $\mathcal{P}$ of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer $n$ and any ordinal $\alpha$ in $\cal P$,thereexistsaseparableBanachspace$X$ such that the Szlenk of the dual of order $k$ of $X$ is equal to the first infinite ordinal $\omega$ for all $k$ in $\{0,\mathrm{.}\mathrm{.},n-1\}$ and equal to $\alpha$ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to $\omega$. We also show that any element of $\mathcal{P}$ can be realized as a Szlenk index of a reflexive Banach space with an unconditional basis.