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Problem 1128 — Erdős Problems

Problem 1128

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: erdosproblems.com/1128

Set theory
Ramsey theory
Hypergraphs

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Solved·, Disproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN).

Er81b: Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35 (page numbers are given for the 2nd edition of The Scottish Book).
Ko25b: P. Komjáth, The Erdős-Hajnal Problem List. Bull. Symb. Log. (2025), 418-461.
To94: Todorčević, Stevo, Some partitions of three-dimensional combinatorial cubes. J. Combin. Theory Ser. A (1994), 410-437.

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Status: Solved, Disproved
Topics: Set theory
Ramsey theory
Hypergraphs
In the wiki: Wiki page