Erdős Problem 1128 (disproved by Prikry–Mills, 1978):

Prikry–Mills counterexample (key lemma):

There exists a 2-colouring $f$ of a set of cardinality $\aleph_1$ cubed such that no countable box $A_1 \times B_1 \times C_1$ is monochromatic.

This is the unpublished result of Prikry and Mills (1978). The proof proceeds by transfinite induction along $\omega_1$, which has uncountable cofinality, ensuring every countable box is non-monochromatic.