Nevertheless ${\mathcal F}\cap[N]^{\omega}$ will be called [*spreading relative to $N$*]{}, if with $A\in  {\mathcal F}\cap[N]^{<\omega}$ every spread of $A$ which is a subset of $N$ is in ${\mathcal F}$. Note that if ${\mathcal F}\subset [N]^{<\omega}$ is spreading relatively to $N=\{n_1,n_2,\ldots \}\in [{\mathbb{N}}]^{\omega}$, then $$\tilde{\mathcal F}=\big\{ A\in[{\mathbb{N}}]^{<\omega}:\{n_j:j\in A\}\in {\mathcal F}\big\} ,$$ is spreading.

A second way to pass to a sub families is the following. Assume that ${\mathcal F}\subset[{\mathbb{N}}]^{<\omega}$ and $N=\{n_1,n_2,\ldots\}\in [{\mathbb{N}}]^{\omega}$, then we call $${\mathcal F}^N=\big\{ \{n_j:j\in A\} : A\in{\mathcal F}\big\},$$ [*the spread of ${\mathcal F}$ onto $N$.*]{}

A family ${\mathcal F}{\!\subset\!}[{\mathbb{N}}]^{<\omega}$ is called [*regular* ]{} if it is hereditary, compact, and spreading. Note that if ${\mathcal F}{\!\subset\!}[{\mathbb{N}}]^{<\omega}$ is compact, spreading and closed under restriction it is also hereditary and thus regular. Indeed if $B=\{b_1,b_2,\ldots, b_l\}\in {\mathcal F}$ and $1\le i_1< i_2<\ldots< i_k\le l $, then $A=\{b_{i_1},b_{i_2}, \ldots, b_{i_k}\}$ is a spread of $B'=\{b_1,b_2,\ldots, b_k\}$ and since $B'\in {\mathcal F}$, it also follows that $A\in{\mathcal F}$.

If ${\mathcal F}{\!\subset\!}[N]^{<\omega}$, for some infinite set $N\subset {\mathbb{N}}$, is called [*regular relatively to $N$*]{}, if it is compact, spreading relatively to $N$ and hereditary.

If ${\mathcal F}\subset [{\mathbb{N}}]^{<\omega}$ we denote the maximal elements of ${\mathcal F}$, [*i.e.,* ]{}the elements $A\in{\mathcal F}$ for which 