TITLE: Minimum cardinality of a cofinal collection of countable subsets of a set
QUESTION [5 upvotes]: Setup

Let $X$ be a set of cardinality $\kappa\geq \aleph_0$.
Edit:
Based on Todd Eisworth's suggestion:
What is the minimum cardinality of a collection $\hat{X}$ of countable subsets of $X$ such that every countable subset $A \subseteq X$ is contained in some element of $\hat{X}$?

REPLY [4 votes]: As it is stated in the comments, the question is about the cofinality of $([\lambda]^{\aleph_0}, \subseteq)$.
The following definition is due to Shelah:
$
cov(\lambda, \mu, \theta, \sigma)=min\{|P|: P$ is a family of subsets of $\lambda$ each of size $< \mu$ such that for every $a \subseteq \lambda, |a|<\theta$, for some $\alpha < \sigma$ and $A_i \in P,$ for $i<\alpha,$ we have $a \subseteq \bigcup_{i<\alpha}A_i   \}$.
It is easily sean that
1-$cf([\lambda]^{\aleph_0}, \subseteq)=cov(\lambda, \aleph_1, \aleph_1, 2)$.
By a theorem of Shelah (see [Sh: 355],  $\aleph _{\omega +1}$ has a Jonsson Algebra, see also Analytic guides and updates for cardinal arithmetic page 23)
2-$\lambda^{\aleph_0}=cov(\lambda, \aleph_1, \aleph_1, 2)+ 2^{\aleph_0}$.
Note that in particular, if $\lambda^{\aleph_0} > 2^{\aleph_0}$, then
$cf([\lambda]^{\aleph_0}, \subseteq)=\lambda^{\aleph_0}$.