Problem 1168

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Statement. Prove that

$$\aleph_{\omega+1}\not\to (\aleph_{\omega+1}, 3,\ldots,3)_{\aleph_0}^2$$

without assuming the generalised continuum hypothesis.

Status. Open.

Source. erdosproblems.com/1168, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1168, https://www.erdosproblems.com/1168.

Formalization. None recorded.

Current assessment

No current assessment is recorded. The status above is imported from the dated site record. This page records no current literature search or independent assessment of proof coverage.

Known Results

Erdős, Hajnal and Rado proved the negative relation under the hypothesis $2^{\aleph_\omega}=\aleph_{\omega+1}$, which the generalized continuum hypothesis implies (Erdős, P., Hajnal, A. and Rado, R., Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93--196; the site's remark credits this paper). Garti, Hayut and Shelah (card; arXiv:2502.16625v2, 25 June 2026, Section 2) force the relation $\aleph_{\omega+1}\not\to(\aleph_{\omega+1},(3)_{\aleph_0})^2$ from a supercompact cardinal together with $\aleph_\omega$ a strong limit and $2^{\aleph_\omega}>\aleph_{\omega+1}$, so the relation is consistent without the local instance of the generalized continuum hypothesis; their Theorem 1.3 obtains the same from a strong cardinal at $\mu=\aleph_{\omega^2}$ in place of $\aleph_\omega$. Whether the relation holds in ZFC, which is what the problem asks, the paper leaves open and says it does not know. The consistency result settles no instance of the question and is not a claim.

Linked library material

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• garti_2025_problem_erdos_hajnal