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

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

Erdős asked whether every 2-colouring of $A \times B \times C$, where $|A| = |B| = |C| = \aleph_1$, must contain a monochromatic countable box $A_1 \times B_1 \times C_1$ with $|A_1| = |B_1| = |C_1| = \aleph_0$.

The answer is No: Prikry and Mills constructed a 2-colouring of $\omega_1^3$ with no monochromatic countable box.

Note: The positive statement asserts that every 2-colouring of every $\aleph_1^3$ contains a monochromatic countably infinite box. Since the answer is False, this positive statement fails.