Problem 1220

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claims/: The 3 claim pages of Problem 1220, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\lambda$ be a singular cardinal such that both $\lambda$ and $\mathrm{cf}(\lambda)$ are $\aleph_0$-inaccessible. Does

$$\lambda \to (\lambda,\aleph_1)^2$$

hold?

Status. Independent. The site's label is OPEN (page last edited 1 September 2026). The not-provable side, that ZFC does not prove the universal statement, is settled by Shelah and Stanley's accepted result [ShSt87, Theorem 3]: in its model $\aleph_{\mathfrak c^+}$, which meets both hypotheses in ZFC by Shelah's bound, fails the relation; Bae's pending claim asserts the same side. The not-disprovable side, that ZFC does not refute the statement, is settled by Erdős, Hajnal and Rado's accepted result [EHR65, Theorem I (iv)], which gives the relation for every qualifying $\lambda$ under GCH, and GCH is consistent with ZFC. One accepted result of each kind settles the question as independent; the frontmatter standing is derived from these two claim pages, and this page departs from the site's label on that ground.

Source. erdosproblems.com/1220, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1220, https://www.erdosproblems.com/1220.

References.

• [EHR65] P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. 16 (1965), no. 1--2, 93--196, doi:10.1007/BF01886396. Library home: erdos_1965_partition_relations_cardinal_numbers.
• [ErHa71] Erdős, P. and Hajnal, A., Unsolved problems in set theory. Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) (1971), 17-48; cited by the site at p. 21.
• [Ko25b] P. Komjáth, The Erdős--Hajnal Problem List. Bull. Symb. Log. 31 (2025), no. 3, 418--461, doi:10.1017/bsl.2025.1; cited by the site at p. 3, where the question is Problem 4. Library home: komjath_2025_erdos_hajnal_problem_list.
• [Sh82] S. Shelah, Proper Forcing. Lecture Notes in Mathematics 940, Springer, 1982; the site's key for Shelah's cardinal-arithmetic theorem, given here as Komjáth's bibliography records the work.
• [ShSt87] S. Shelah and L. J. Stanley, A theorem and some consistency results in partition calculus. Ann. Pure Appl. Logic 36 (1987), no. 2, 119--152, doi:10.1016/0168-0072(87)90015-7; Shelah archive Sh:258, whose copy is the published version.
• [ShSt93] S. Shelah and L. Stanley, More consistency results in partition calculus. Israel J. Math. 81 (1993), 97--110.

Formalization. None recorded by the site or the community database, and the formal-conjectures repository has no statement file for 1220. The claimant's own Lean 4 development is linked on Bae's claim page; nothing was built here.

Current assessment

Scope: the site's problem page, proof-claims tab and commentary were read; Komjáth's commentary on Problem 4 of the list [Ko25b, pp. 419--420] was read in the edition the library card names; [ShSt87] was read in the Shelah archive's copy for its list of results (p. 119), its discussion and historical remarks (pp. 122--126), the first and last pages of §3 (pp. 139 and 144) and its references (p. 152), and the proof of its Theorem 3 was not followed; [EHR65] was read on page images of a scan of the published paper for its conventions (pp. 96--97), Theorem I with its critical number (p. 130) and the proof of its part (iv) (p. 135), with the statements that this proof applies, Corollary 1 (p. 105), Theorem 5 (p. 108) and Lemma 4 (pp. 113--114), whose proofs were not followed; the claimant's manuscript, repository README and registry entry were read as the claimant posts them, and the claim is assessed below. No literature search beyond the site, Komjáth's survey, [ShSt87] and [EHR65] and no independent assessment of proof coverage is recorded; [ShSt93], the book of Erdős, Hajnal, Máté and Rado and Hajnal and Larson's Handbook chapter have not been read at source level here.

A positive answer under GCH is classical. Erdős, Hajnal and Rado note, as the site's commentary records, that the answer is positive under GCH or under other additional assumptions on $\lambda$, and their first Main Theorem gives it: [EHR65, Theorem I (iv)] (stated p. 130, proved in §15.8, p. 135, under GCH) gives $\lambda\to(\lambda,\aleph_1)^2$ for singular $\lambda$ whenever $\aleph_1<\mathrm{cf}(\lambda)$ and, if $\mathrm{cf}(\lambda)=\nu^+$, $\mathrm{cf}(\nu)>\aleph_0$; both conditions follow from the $\aleph_0$-inaccessibility of $\mathrm{cf}(\lambda)$, by König's theorem in the successor case. Its proof passes to the cofinality by the paper's Lemma 4 (p. 113), which under GCH makes $\lambda\to(\lambda,\aleph_1)^2$ equivalent to $\mathrm{cf}(\lambda)\to(\mathrm{cf}(\lambda),\aleph_1)^2$. Komjáth [Ko25b, p. 419] cites the Erdős--Rado theorem, as [45, Theorem 35.4] of the survey's bibliography, for the positive relation at $\aleph_{\mathfrak c^+}$ when that cardinal is strong limit. Since GCH holds in Gödel's constructible universe, the universal statement is consistent with ZFC, which is the not-disprovable side, accepted on Erdős, Hajnal and Rado's claim page.

Komjáth's commentary on Problem 4 [Ko25b, pp. 419--420] reports that $\aleph_{\mathfrak c^+}$ satisfies both hypotheses in ZFC, by Shelah's theorem [Sh82], that Shelah and Stanley [ShSt87] force a counterexample to $\aleph_{\mathfrak c^+}\to(\aleph_{\mathfrak c^+},\aleph_1)^2$, and that [ShSt93] gives, from $\mathfrak c^+$ measurable cardinals, a model in which $\aleph_{\mathfrak c^+}$ is not strong limit yet the relation holds. The first two reports agree with [ShSt87]: its Theorem 3 (stated p. 119, proved in §3, pp. 139--144) reads that if ZFC is consistent, then so is ZFC + $\aleph_{\mathfrak c^+}\not\to(\aleph_{\mathfrak c^+},\aleph_1)^2$, and its historical remarks (p. 125) record Shelah's ZFC bound $\lambda^{\aleph_0}<\aleph_{\mathfrak c^+}$ for $\lambda<\aleph_{\mathfrak c^+}$ (cited there from a 1980 note on cardinal exponentiation) and the theorem of Erdős, Hajnal, Máté and Rado that the positive relation at $\aleph_{\mathfrak c^+}$ holds unless $2^\lambda>\aleph_{\mathfrak c^+}$ for some $\lambda<\aleph_{\mathfrak c^+}$. The paper treats the instance at $\aleph_{\mathfrak c^+}$, which it traces to Problem 35.5 of their book, and draws no conclusion about the universal question; its Theorem 6 (p. 119), deferred to [ShSt93], is the positive relation at a non-strong-limit $\aleph_{\mathfrak c^+}$ from $\mathfrak c^+$ measurable cardinals, an instance only. So [ShSt87] gives the not-provable side, on Shelah and Stanley's claim page, and [EHR65] the not-disprovable side; both are accepted on their refereed publication, and one accepted result of each kind settles the question as independent.

Claims. Three results are claimed from outside the project. Two are accepted on their refereed publication and settle one side each: Shelah and Stanley's consistency result (Ann. Pure Appl. Logic, 1987) the not-provable side and Erdős, Hajnal and Rado's theorem under GCH (Acta Math. Acad. Sci. Hungar., 1965) the not-disprovable side; together they derive the standing solved with claim independent. Bae's non-provability claim (Zenodo manuscript of 2026-09-25, Lean 4 development registered on the Palomar registry on 2026-09-26, posted on the site's proof-claims tab the same day) asserts that ZFC, if consistent, does not prove the universal statement, through the Shelah--Stanley forcing [ShSt87, Theorem 3] at $\lambda=\aleph_{\mathfrak c^+}$. The manuscript shows the instance at that cardinal independent of ZFC, since the relation holds there under GCH. The claim page records the claimant's assertion, non-provability, as a partial claim: it covers the not-provable side, and one side alone leaves the question open. The registry replays the proof in the Lean kernel and compares it with a challenge statement; by its own description it certifies neither novelty nor the match between the formal and informal statements and is not peer review. No refereed version, site acceptance or independent review is recorded, the registered statement was not read and nothing was built here, so the claim stays claimed. It asserts the side that Shelah and Stanley's accepted page settles, and the derived standing does not rest on it; the site shows OPEN and the community database says open.

Known Results

The two sides of the independence are on the accepted claim pages: the consistency of $\aleph_{\mathfrak c^+}\not\to(\aleph_{\mathfrak c^+},\aleph_1)^2$ [ShSt87, Theorem 3] on Shelah and Stanley's page, and the relation for every qualifying $\lambda$ under GCH [EHR65, Theorem I (iv)] on Erdős, Hajnal and Rado's page. The claimed non-provability of the universal statement is on Bae's claim page; Komjáth [Ko25b, p. 419] cites the Erdős--Rado theorem for the relation at $\aleph_{\mathfrak c^+}$ when that cardinal is strong limit; no other result is 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