Problem 1169

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claims/: The 5 claim pages of Problem 1169, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that, for all finite $k<\omega$,

$$\omega_1^2 \not\to (\omega_1^2, 3)^2?$$

Status. Open. The site labels the problem NOT DISPROVABLE and credits Hajnal's proof of the negative relation under the continuum hypothesis. That consistency result, with the later ones from a Suslin tree, from the stick principle, from $\mathfrak d=\aleph_1$ and in a model of a fragment of Martin's axiom, shows that ZFC does not refute the relation, one side of an independence result. Whether ZFC proves it, that is, whether $\omega_1^2\to(\omega_1^2,3)^2$ is consistent, is open. This page departs from the site's label because one side alone leaves the question open.

Source. erdosproblems.com/1169, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1169, https://www.erdosproblems.com/1169.

References.

• [Ha71] Hajnal, A., A negative partition relation. Proc. Nat. Acad. Sci. U.S.A. (1971), 142-144.
• [Va99] Some of Paul's favorite problems, booklet for the conference "Paul Erdős and his mathematics", Budapest, July 1999; item 7.85, which asks whether $\omega_1^2\not\to(\omega_1^2,3)^2$ and notes that Hajnal derived it from CH. Library home: various_1999_some_pauls_favorite_problems.

Formalization. None recorded.

Current assessment

The statement above, as the site gives it, is the one whose standing is recorded. Its quantifier "for all finite $k<\omega$" binds nothing in the displayed relation; the booklet item [Va99, 7.85] asks the same question without it, and at $k=0$ a reading with $k$ triangle colors is the false relation $\omega_1^2\not\to(\omega_1^2)^2_1$; the displayed relation without $k$ is the one whose standing is recorded. Hajnal [Ha71] proves $\omega_1^2\not\to(\omega_1^2,3)^2$ from the continuum hypothesis, so ZFC does not refute the relation; the claim page Hajnal 1971 carries the statement, its source and the acceptance evidence, a refereed paper and the site's curator crediting it. Whether the relation is a theorem of ZFC, that is, whether $\omega_1^2\to(\omega_1^2,3)^2$ is consistent, is open: Komjáth's 2025 survey says so in its Problem 13 commentary (source card). The negative relation does not need CH. It follows from a Suslin tree (Baumgartner 1975), from the stick principle (Takahashi 1987) and from $\mathfrak d=\aleph_1$ (Larson 1998), each consistent with the failure of CH. A 2026 preprint (Golshani, arXiv:2608.13213) states it in a model of $\mathrm{MA}_{\omega_1}(\sigma\text{-centered})$ with $2^{\aleph_0}=\aleph_2$. No model of the positive relation is known. Each of these results settles the same side, that ZFC does not refute the relation, so each claim page is partial and the problem is open: it would be settled as independent by a model of $\omega_1^2\to(\omega_1^2,3)^2$, and as proved by a proof of the negative relation in ZFC alone. No literature search beyond these sources and the site is recorded, and nothing on this page is independently reviewed by this project.

Known Results

The Current assessment above records the known results.

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.

• various_1999_some_pauls_favorite_problems
• erdos_1974_unsolved_solved_problems_set_theory
• erdos_1974_unsolved_solved_problems_set_theory / theorem_p273_hajnal
• komjath_1988_forcing_constructions_uncountably_chromatic_graphs
• komjath_1988_forcing_constructions_uncountably_chromatic_graphs / theorem_7
• komjath_2025_erdos_hajnal_problem_list