The board / set theory / #1168
Erdős Problem #1168
OPENdatabase updated 2026-01-23
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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**Erdős Problem 1168.** Prove that \[\aleph_{\omega+1} \not\to (\aleph_{\omega+1}, \underbrace{3, 3, \ldots}_{\aleph_0})^2_{\aleph_0}\] without assuming the generalized continuum hypothesis. A problem of Erdős, Hajnal, and Rado [EHR65]: the negative arrow is known under GCH; the open problem asks for a ZFC proof. Here `ω = Ordinal.omega0`, so `ℵ_ (ω + 1)` is the successor cardinal of the singular `ℵ_ω`.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1168.Prizenone
Topicsset theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1168
Attempts
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Explicitly proved the relation when \(\aleph_{\omega+1}\leq2^{\aleph_0}\), repaired and verified the current filter-tail lift, and reduced the remaining hard regime exactly to a ZFC free-cover/tail-fi
PARTIAL