Help with a problem about consequences of the continuum hypothesis

Question

Suppose that the continuum hypothesis holds. I'm trying to prove that there is a set $T\subseteq\omega_1\times\omega$ such that every set $S$ with $S=A\times B\subseteq\omega_1\times\omega$ with $A$ being uncountable and $B$ being infinite meets both $T$ and $T$'s complement in $\omega_1\times\omega$. Not really sure how to begin: I don't see how the continuum hypothesis could be of any help here.

Answer 1

Let $\langle X_i : i < \omega_1 \rangle$ list all infinite subsets of $\omega$. For each $i < \omega_1$, choose $Y_i \subseteq \omega$ such that for every $j < i$, both $Y_i \cap X_j$ and $(\omega \setminus Y_i) \cap X_j$ are infinite. Put $T = \{(i, n) : i < \omega_1, n \in Y_i\}$. Now check.