and the [*Fine Schreier Families* ]{} ${\mathcal F}_{\beta,\alpha}$, $\alpha\le \beta<\omega_1$ [@AA]. The recursive definition of both families depends on choosing for every limit ordinal a sequence $(\alpha_n)$, which increases to $\alpha$. In order to ensure that later our proof will work out, we will need $(\alpha_n)$ to satisfy certain conditions.

Regular Families in $[{\mathbb{N}}]^{<\omega}$ {#SS:2.1}
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For a set $S$ we denote the subsets, the finite subsets, and the countably infinite subsets by $[S]$, $[S]^{<\omega}$, and $[S]^\omega$. We always write subsets of ${\mathbb{N}}$ in increasing order. Thus, if we write $A=\{a_1,a_2,\ldots a_n\}\in[{\mathbb{N}}]^{<\omega}$, or $A=\{a_1,a_2,\ldots \}{\!\in\!}[{\mathbb{N}}]^{\omega}$ we always assume that $a_1{\!<\!}a_2{\!<\!}\ldots $. Identifying the elements of $[{\mathbb{N}}]$ in the usual way with elements of $\{0,1\}^\omega$, we consider on $[{\mathbb{N}}]$ the product topology of the discrete topology on $\{0,1\}$. Note that it follows for a sequence $(A_n){\!\subset\!}[{\mathbb{N}}]^{<\omega}$ and $A{\!\in\!}[{\mathbb{N}}]^{<\omega}$, that $(A_n)$ converges to $A$ if and only if for all $k{\!\ge\!}\max A$ there is an $m$ so that $A_n\cap[1,k]{\!=\!}A$, for all $n{\!\ge\!}m$.

For $A{\!\in\!}[{\mathbb{N}}]^{<\omega}$ and $B{\!\in\!}[{\mathbb{N}}]$ we write $A{\!<\!}B$, if $\max(A){\!<\!}\min(B)$. As a matter of convention we put $\max(\emptyset)=0$ and $\min(\emptyset)=\infty$, and thus $A{\!<\!}\emptyset$ and $\emptyset{\!>\!}A$ is true for all $A \in [{\mathbb{N}}]^{<\omega}$. For $m\in{\mathbb{N}}$ we write $m\le A$ or $m<A$, if 