The background theory here is at least ZFC and probably ZFC+MA+$\neg$CH, if it matters. Here's the question:

Suppose that $C\subseteq\omega_2\times\omega_2$. Must there exist uncountable sets $A,B\subseteq\omega_2$ such that either $A\times B\subseteq C$ or $(A\times B)\cap C=\emptyset$?

Note that the usual issue with coloring *ordered* pairs, namely coloring them based on whether they agree with underlying ordering, doesn't present itself here since I'm not asking for $A=B$ or $|A|=|B|=\aleph_2$.

My other thought for showing this to be false would be to do a kind of "Bernstein-like" construction, enumerating all pairs $A,B\subseteq\omega_2$ of size $\aleph_1$ and diagonalizing against the products $A\times B$. Assuming MA+$2^{\aleph_0}=\aleph_2$, we'd have that $|\mathcal{P}_{\aleph_1}(\omega_2)|=\aleph_2^{\aleph_1}=2^{\aleph_1}=2^{\aleph_0}=\aleph_2$, but the construction breaks down once $\geq\aleph_1$ many points have been chosen.

If this statement turns out to be true, I'd also be interested in higher dimensional forms (sets of n-tuples, etc).

**Edit:** After looking around some more, I think what I am really asking is whether the polarized partition relation $\binom{\aleph_2}{\aleph_2}\to\binom{\aleph_1}{\aleph_1}^{1,1}$ (just a restatement of my question above) is consistent with MA+$\neg$CH. Notably, Hajnal proved this relation under GCH.