Concerning the cardinality of a collection of subsets of the natural numbers

Question

Let $S$ be a collection of subsets of the natural numbers $\mathbb N$ with the following property: If $A$ is an infinite subset of $\mathbb N$ then there is an infinite subset $B$ of $A$ for which $B$ is an element of $S$. Are there models of set theory in which the minimum cardinality of $S$ is less than the cardinality of the continuum?

Answer 1

No. This will always be the size of the continuum.

To see this, recall that there is an almost disjoint family of size $2^{\aleph_0}$, where almost disjoint means that the intersection of any two distinct members of the family is finite.

So, in that case, any infinite subset of a member of the family can be matched to the unique one which contains it (other sets might contain finite parts of it, of course, but not the entire set).