a separable and reflexive space $X$

1.  $\max({\text{\rm Sz}}(X),{\text{\rm Sz}}(X^*))\le \alpha$,

2.  $({\mathcal S}_\alpha,d_1)$ is not bi-Lipschitzly embeddable into $X$.

Corollary \[C:1.1\] and a result in [@OSZ1] yield the following corollary. We thank Christian Rosendal who pointed it out to us.

\[C:1.2\] If $\alpha{\!<\!}\omega_1$, with $\alpha=\omega^\alpha$, the class of all Banach spaces $X$ for which $\max({\text{\rm Sz}}(X),{\text{\rm Sz}}(X^*)){\!\leq\!}\alpha$ is Borel in the Effros-Borel structure of closed subspaces of $C[0,1]$.

A proof of Corollaries \[C:1.1\] and \[C:1.2\] will be presented at the end of Section \[S:12\]. For the proof of our main results we will need to introduce some notation and to make several preliminary observations. The reader who is at first only interested to understand our main results will only need the definition of the [*Schreier families*]{} ${\mathcal S}_\alpha$, $\alpha<\omega_1$, given in Subsection \[SS:2.2\], the definition of [*repeated averages*]{} stated in the beginning of of Section \[S:4\], and the definition of the two metrics $d_{1,\alpha}$ and $d_{\infty,\alpha}$ on ${\mathcal S}_\alpha$ introduced in Section \[S:8\].

Regular Families, Schreier Families and Fine Schreier Families {#S:2}
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In this section we will first recall the definition of general [*Regular Subfamilies of* ]{}$[{\mathbb{N}}]^{<\omega}$. Then we recall the definition of the [*Schreier Families*]{} ${\mathcal S}_\alpha$ 