Problem 1220

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?

Source: erdosproblems.com/1220

Set theory
Ramsey theory

Wiki page

Status

SolvedIndependent

An accepted solution exists. The statement can be neither proved nor disproved from the standard axioms of set theory.

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.

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: Erdős and Rado 1965.
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: Komjáth 2025.
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.

Problem 1220

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