definition of the [*Schreier families ${\mathcal S}_\alpha\subset[{\mathbb{N}}]^{<\omega}$,*]{} for $\alpha<\omega_1$, will be recalled in Section \[S:2\], the [*Szlenk index* ]{}${\text{\rm Sz}}(X)$ for a Banach space $X$ in Section \[S:6\], and the two metrics $d_{1,\alpha}$ and $d_{\infty,\alpha}$ on ${\mathcal S}_\alpha$ will be defined in Section \[S:8\]. The statements of our main results are as follows.

A separable Banach space $X$ is reflexive if and only if there is an $\alpha<\omega_1$ for which there does not exist a map $\Phi: {\mathcal S}_\alpha\to X$, with the property that for some numbers $C\ge c>0$ $$\label{E:1.1}
    cd_{\infty,\alpha}(A,B)\le \|\Phi(A)-\Phi(B)\|\le C  d_{1,\alpha}(A,B) \text{ for all $A,B\in{\mathcal S}_\alpha$.}$$

Assume that $X$ is a Banach space, $\alpha<\omega_1$, and $C\ge c>0$. We call a map $\Phi: {\mathcal S}_\alpha\to X$, with the property that for all $A,B\in{\mathcal S}_\alpha$, $$\label{E:1.2}
    cd_{\infty,\alpha}(A,B)\le \|\Phi(A)-\Phi(B)\|\le C  d_{1,\alpha}(A,B)$$ a [*$c$-lower-$d_{\alpha,\infty}$ and $C$-upper-$d_{\alpha,1}$ embedding of ${\mathcal S}_\alpha$ into $X$*]{}. If ${\mathcal A}$ is a subset of ${\mathcal S}_\alpha$, and $\Phi:{\mathcal A}\to X$, is a map which satisfies for all $A,B\in {\mathcal A}$, we call it a [*$c$-lower-$d_{\alpha,\infty}$ and $C$-upper-$d_{\alpha,1}$ embedding of ${\mathcal A}$ into $X$*]{}.

Our next result extends one direction (the “easy direction”) of the main result of [@BKL] to spaces with higher order Szlenk indices. As 