A Ramsey-type result for families of subsets

Question

Let $S$ be a set of cardinality $\aleph_1$. Consider the directed family $\mathcal{C}$ (here directed means directed with respect to the inclusion) of all countably infinite subsets of $S$. Suppose that

$$\mathcal{C} = \bigcup_{n=1}^\infty \mathcal{C}_n$$

for some families $\mathcal{C}_n$. Does it follow that for some $n_0$ the family $\mathcal{C}_{n_0}$ contains an uncountable, directed subfamily?

Of course, at least one $\mathcal{C}_n$ is uncountable, so let us take this one. Must it contain an uncountable directed subfamily?

Answer 1

Yes. Letting $S = \omega_1$, one of the $C_n$'s must contain uncountably many ordinals and hence an uncountable linearly ordered subfamily.

Also not every uncountable family contains one. For example, an almost disjoint family.

Answer 2

Improving on hot-queens result, note that this is true even if S is countably infinite. To show this identify S with rationals and note that some $C_n$ must contain uncountably many reals where we view a real x as the set of rationals less than x.