$m\le \min(A)$, or $m<\min(A)$, respectively.

We denote by $\preceq$ the partial order of [*extension*]{} on $[{\mathbb{N}}]^{<\omega}$, [*i.e.,* ]{}$A=\{a_1,a_2,\ldots a_l\}\preceq B=\{b_1,b_2,\ldots b_m\} $ if $l\le m$ and $a_i=b_i$, for $i=1,2\ldots l$, and we write $A\prec B$ if $A\preceq B$ and $A\not= B$.

We say that ${\mathcal F}{\!\subset\!}[{\mathbb{N}}]^{<\omega}$ is [*closed under taking restrictions*]{} or [*is a subtree of $[{\mathbb{N}}]^{<\omega}$*]{} if $A{\!\in\!}{\mathcal F}$ whenever $A\prec B$ and $B{\!\in\!}{\mathcal F}$, *hereditary* if $A{\!\in\!}{\mathcal F}$ whenever $A{\!\subset\!}B$ and $B{\!\in\!}{\mathcal F}$, and ${\mathcal F}$ is called *compact* if it is compact in the product topology. Note that a family which is closed under restrictions is compact if and only if it is [*well founded*]{}, [*i.e.,* ]{}if it does not contain strictly ascending chains with respect to extensions. Given $n,\ a_1{\!<\!}\dots{\!<\!}a_n,\ b_1{\!<\!}\dots{\!<\!}b_n$ in ${\mathbb{N}}$ we say that $\{b_1,\dots,b_n\}$ is *a spread* of $\{a_1,\dots,a_n\}$ if $a_i{\!\leq\!}b_i$ for $i{\!=\!}1,\dots,n$. A family ${\mathcal F}{\!\subset\!}[{\mathbb{N}}]^{<\omega}$ is called *spreading* if every spread of every element of ${\mathcal F}$ is also in ${\mathcal F}$. We sometimes have to pass from a family ${\mathcal F}\subset [{\mathbb{N}}]^{<\omega}$ to the subfamily ${\mathcal F}\cap[N]^{\omega}= \{ A\in {\mathcal F}: A\subset N\}$, where $N\subset {\mathbb{N}}$ is infinite. Of course we might lose the property that ${\mathcal F}$ is spreading. 