Erdős Problem #1128

- $50

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?

#1128: [Er81b,p.33]

A problem of Erdős and Hajnal.

This was disproved by Prikry and Mills in 1978 but this seems to have been unpublished. This is reported by Todorčević [To94] and Komjáth [Ko25b].

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This page was last edited 30 December 2025.

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