Open. The site labels the problem NOT DISPROVABLE and credits Hajnal's proof of the negative relation under the continuum hypothesis. That consistency result, with the later ones from a Suslin tree, from the stick principle, from $\mathfrak d=\aleph_1$ and in a model of a fragment of Martin's axiom, shows that ZFC does not refute the relation, one side of an independence result. Whether ZFC proves it, that is, whether $\omega_1^2\to(\omega_1^2,3)^2$ is consistent, is open. This page departs from the site's label because one side alone leaves the question open.