The board / set theory / #1128
Erdős Problem #1128
DISPROVED LEANdatabase updated 2026-08-23
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A subset \(A_1 \times B_1 \times C_1\) of \(A \times B \times C\) is **monochromatic** under a 2-colouring \(f : A \to B \to C \to \operatorname{Fin} 2\) if \(f\) is constant on \(A_1 \times B_1 \times C_1\).
The set of countable ordinals, expressed using Mathlib's `ω_ 1`.
**Prikry–Mills counterexample** (key lemma): There exists a 2-colouring \(f\) of a set of cardinality \(\aleph_1\) cubed such that no countable box \(A_1 \times B_1 \times C_1\) is monochromatic. This is the unpublished result of Prikry and Mills (1978). The proof proceeds by transfinite induction along \(\omega_1\), which has uncountable cofinality, ensuring every countable box is non-monochromatic.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1128.Prize$50
Topicsset theory, ramsey theory, hypergraphs
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1128
Attempts
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