Problem 1128

../

claims/: The 1 claim page of Problem 1128, one per claimant's result; the problem's standing derives from them.

---

Statement. 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?

Status. DISPROVED (LEAN).

Source. erdosproblems.com/1128, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1128, https://www.erdosproblems.com/1128.

References.

• [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.

Formalization. The formal-conjectures statement file, at the commit linked, states the problem with the answer answer(False), is marked research solved, and attributes the counterexample to Prikry and Mills, but leaves its construction as sorry. A Lean proof of the result is in Boris Alexeev's lean-proofs repository, with Codex and GPT-5.6 Sol named as its formal authors; the claim page links it at its pinned commit and records both files. The corpus has built neither file.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• komjath_2025_erdos_hajnal_problem_list