Erdős problem 1128

The claim that every 2-colouring of $\omega_1 \times \omega_1$ has an uncountable monochromatic product rectangle is false in ZFC.

Counterexample: The ordering colouring $f(\alpha, \beta) = 0$ iff $\alpha < \beta$ has no uncountable monochromatic product rectangle $A_1 \times B_1$.

Proof: If $A_1 \times B_1$ were monochromatic with colour 0, then every element of $A_1$ would be strictly less than every element of $B_1$, making $A_1$ bounded above in $\omega_1$; but any bounded subset of $\omega_1$ is countable (since initial segments are countable), contradicting $A_1$ being uncountable. The colour-1 case is symmetric with the roles of $A_1$ and $B_1$ swapped.

Note: The correct classical result for 2-colourings of pairs (not products) is the Erdős–Rado theorem $\omega_1 \to (\omega_1)^2_2$, which concerns unordered pairs.