[{\mathbb{N}}]$ we define $$\begin{aligned}
\label{E:2.1.1}
&{\mathcal F}\sqcup_< {\mathcal G}:=\big\{A\cup B: A\in {\mathcal F},\,\,B\in {\mathcal G}\text{ and } A<B \big\}\\
\label{E:2.1.2}&{\mathcal F}[{\mathcal G}]:=\left\{ \bigcup_{i=1}^n B_i: \begin{matrix} n\in{\mathbb{N}},\, B_1<B_2<\ldots< B_n,\, B_i\in{\mathcal G},\, i=1,2,\ldots n,\\
         \text{ and } \{\min(B_i):i=1,2\ldots n\}\in {\mathcal F}\end{matrix}\right\}.\end{aligned}$$

It is not hard to see that if ${\mathcal F}$ and ${\mathcal G}$ are regular families so are ${\mathcal F}\sqcup_<{\mathcal G}$ and ${\mathcal F}[{\mathcal G}]$.

The Schreier families {#SS:2.2}
---------------------

We define the [*Schreier families*]{} ${\mathcal S}_\alpha\subset[{\mathbb{N}}]^{<\omega}$ by transfinite induction for all $\alpha<\omega_1$, as follows : $$\begin{aligned}
\label{E:2.3}
&{\mathcal S}_0=\big\{ \{n\}: n\in{\mathbb{N}}\big\}\cup\{\emptyset\}
\intertext{if $\alpha=\gamma+1$, we let }
\label{E:2.4}&{\mathcal S}_{\alpha}={\mathcal S}_1[{\mathcal S}_\gamma]
    =\Big\{ \bigcup_{j=1}^n E_j: n{\!\leq\!}\min(E_1),\, E_1< E_2<\ldots< E_n,\, E_j{\!\in\!}{\mathcal S}_{\gamma}, j=1,2,\ldots,n \Big\},\\
\intertext{and if $\alpha$ is a limit ordinal we choose a fixed sequence $\big(\lambda(\alpha,n):n\in{\mathbb{N}}\big)\subset[1,\alpha)$ which increases to $\alpha$ and put}
 \label{E:2.5}  &{\mathcal S}_\alpha=\{  E: \exists \, k\le \min(E), \text{ with $E{\!\in\!}{\mathcal S}_{\lambda(\alpha,k)}$} \}.
 \end{aligned}$$

An easy induction shows that ${\mathcal S}_\alpha$ is a hereditary, compact and spreading family for all $\alpha{\!<\!}\omega_1$. It is not very hard to see by transfinite induction that ${\mathcal S}_\alpha$ is in the following very limited sense [*backwards spreading*]{}: $$\begin{aligned}
\label{E:2.6}  &\text{If $A=\{a_1,a_2, \ldots,a_n\}\in{\mathcal S}_\alpha$ , then }
  \text{$\{a_1,a_2,\ldots,a_{n-1},a_{n}-1\} \in S_\alpha$.}\end{aligned}$$ So, in particular, if $A\in{\mathcal 