Problem 1218

Assume the Generalised Continuum Hypothesis and let $\beta=\omega_{\omega+1}$ . Is it true that

\aleph_{\beta+1} \not\to (\aleph_\beta,4,\ldots,4)^3

(where there are countably infinite many copies of $4$ )?

Source: erdosproblems.com/1218

Set theory
Ramsey theory

Wiki page

Status

Open

No claim settles this problem.

Open. The site's label is OPEN (page last edited 7 September 2026, as of 2026-10-07). The one result claimed against the problem addresses the site's earlier wording and is rejected on its claim page; the standing in the frontmatter is derived from the claim pages.

EHR65: Erdős, P., Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. 16 (1965), no. 1--2, 93--196, doi:10.1007/BF01886396; cited by the site at p. 131. Library home: Erdős and Rado 1965; the arrows of Corollary 13 (p. 138) are negated, as part (ii) of Theorem II, which says the relation is false, supplies.
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. 20.
Ko25b: P. Komjáth, The Erdős--Hajnal Problem List. Bull. Symb. Log. 31 (2025), no. 3, 418--461, doi:10.1017/bsl.2025.1 (the site's bibliography prints "Probem"); cited by the site at p. 2. Library home: Komjáth 2025.

Problem 1218

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Proof claims

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