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For $A, B\subseteq \omega$ we write $A\subseteq^*B$ if $A\setminus B$ is finite. Let $[\omega]^\omega$ be the collection of infinite subsets.

We say that ${\cal D}\subseteq [\omega]^\omega$ is dominating from below if for all $X\in [\omega]^\omega$ there is $D\in {\cal D}$ with $D \subseteq^* X$. Let ${\frak D}$ be the collection of families in $[\omega]^\omega$ that are dominating from below.

Question. Is it consistent that $\min\{|{\cal D}|: {\cal D} \in {\frak D}\} < 2^{\aleph_0}$?

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    $\begingroup$ Do you mean to write $A \subseteq^* B$ if rather $A \setminus B$ is finite? Or perhaps $A \subseteq B$ and $B \setminus A$ is finite? Otherwise $\{\omega\}$ is dominating from below. $\endgroup$ Commented Mar 27, 2025 at 10:35
  • $\begingroup$ Thanks for noticing my error - will correct it! $\endgroup$ Commented Mar 27, 2025 at 12:37

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The answer is no; this cardinal is always $2^\omega$.

Using the tree $2^{<\omega}$, it is easy to build an almost-disjoint collection $\mathcal A$ of infinite subsets of $\omega$ of size continuum. If you have a set $\mathcal D$ as in your hypothesis, then for all $A \in \mathcal A$, there is $D_A \in \mathcal D$ that is almost contained in $A$. Since $\mathcal A$ is almost disjoint, the function $A \mapsto D_A$ is injective.

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