Problem 1171

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claims/: The 3 claim pages of Problem 1171, one per claimant's result; the problem's standing derives from them.

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

$$\omega_1^2\to (\omega_1\omega, 3,\ldots,3)_{k+1}^2?$$

Status. Open. The site labels the problem NOT DISPROVABLE, and its label note says the question is open in general and holds in some models of set theory. The accepted partial claim Baumgartner's partition ordinal under Martin's axiom shows that ZFC does not refute the relation, one side of an independence result, while whether ZFC proves it is open for $k\ge3$. The ZFC cases $k\le2$ are on the claimed partial page Baumgartner and Hajnal 1987; the deposit that prompted the site's label has its own page, Gao's conditional proof, recorded as withdrawn.

Source. erdosproblems.com/1171, accessed 2026-09-27: the problem page (NOT DISPROVABLE; its label note says the question is open in general and holds in some models of set theory; last edited 26 January 2026; source key [Va99, 7.84]; remark citing [Ba89b] for $\omega_1\omega\to(\omega_1\omega,3)^2$ under a form of Martin's axiom), its one-comment discussion thread (5 September 2026), its proof-claims tab, which listed one partial claim (submitted 2026-09-05) and no longer lists it, and its revision history, which shows the statement unchanged. Cite as: T. F. Bloom, Erdős Problem #1171, https://www.erdosproblems.com/1171.

References.

• [Ba89b] Baumgartner, James E., Remarks on partition ordinals. In: Set theory and its applications (Toronto, ON, 1987), Lecture Notes in Math. 1401, Springer, Berlin, 1989, pp. 5-17. DOI 10.1007/BFb0097328. Not held (paywalled); statement from its zbMATH review. Library home: baumgartner_1989_remarks_partition_ordinals.
• [BaHa87] Baumgartner, James E. and Hajnal, András, A remark on partition relations for infinite ordinals with an application to finite combinatorics. Contemp. Math. 65 (1987), 157-167. DOI 10.1090/conm/065/891246. Not held (paywalled); statements from its zbMATH review and from [Ko25]. Library home: baumgartner_1987_remark_partition_relations_infinite_ordinals.
• [ErHa70] Erdős, P. and Hajnal, A., Some results and problems on certain polarized partitions. Acta Math. Acad. Sci. Hungar. 21 (1970), 369-392. Open scan at https://users.renyi.hu/~p_erdos/1970-25.pdf; not held. Komjáth's reference [40], which renders the title with "for" in place of "on", is his source for attributing the case $k=1$ to this paper; the attribution is Komjáth's and is not found in the paper, whose §1 says it does not investigate relations for order types and gives its definitions for cardinals only.
• [ErHa71] Erdős, P. and Hajnal, A., Ordinary partition relations for ordinal numbers. Period. Math. Hungar. 1 (1971), no. 3, 171-185. DOI 10.1007/BF02029142. Treats the ordinal relation $\omega_1^2\to(\mu,3)^2$ for $\mu<\omega_1^2$, which its zbMATH review (Zbl 0257.04004) states under GCH; not held, and not established to prove the case $k=1$ in ZFC.
• [Ga26] Gao, Lezhe, A finite-color partition relation for $\omega_1^2$ under $\mathrm{MA}_{\aleph_1}$. Zenodo (5 September 2026), DOI 10.5281/zenodo.22315956. Unrefereed, AI-assisted; withdrawn by its author on 1 October 2026 (the Zenodo record is a tombstone giving retraction or withdrawal as the reason). Library home: gao_2026_finite_color_partition_relation_omega_1_squared.
• [Ko25] Komjáth, Péter, The Erdős–Hajnal problem list. Bull. Symbolic Logic 31 (2025), 418-461. DOI 10.1017/bsl.2025.1. Library home: komjath_2025_erdos_hajnal_problem_list.
• [SoTe71] Solovay, R. M. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem. Ann. of Math. (2) 94 (1971), 201-245. The relative consistency of Martin's axiom with the negation of CH; not held.
• [Va99] Some of Paul's favorite problems, booklet for the conference "Paul Erdős and his mathematics", Budapest, July 1999; item 7.84, which asks whether $\omega_1^2\to(\omega_1\omega,(3)_k)^2$ for every $k<\omega$ (section 7). Library home: various_1999_some_pauls_favorite_problems.

Formalization. None recorded. The site answers its formalized-statement field with "No"; no file ErdosProblems/1171.lean exists in google-deepmind/formal-conjectures (the path returns HTTP 404 while sibling files resolve); conjectures.io lists no settled item and no problem for 1171; and the community database records the formal status as unformalized (all checked 2026-09-27).

Current assessment

The dated site formulation (erdosproblems.com/1171, last edited 26 January 2026, accessed 2026-09-27) asks whether

$$\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$$

holds for every finite $k$, with $k$ triangle targets: every coloring of the pairs from $\omega_1^2$ with $k+1$ colors has a set of order type $\omega_1\omega$ homogeneous in color $0$ or a triangle homogeneous in one of the other colors. The booklet item [Va99, 7.84] asks the same question in the notation $(\omega_1\omega,(3)_k)^2$. The problem is open. The relation is not known to be a theorem of ZFC, but it holds in every model of $\mathrm{MA}_{\aleph_1}$, Martin's axiom for $\aleph_1$ dense sets, which is consistent relative to ZFC [SoTe71], so ZFC does not refute it. That is one side of an independence result; the other side, a model in which the relation fails, is not known for $k\ge3$, and for $k\le2$ the relation is a theorem of ZFC. The site labels the problem not disprovable: its label and the community database changed from open to not disprovable on 5 September 2026, after a comment in the discussion thread asked for the change on the strength of the deposit [Ga26]. The author withdrew the deposit on 1 October 2026 and the site's proof-claims tab no longer lists it, while the label stands; the claim pages record both.

Consistency. The consistency rests on Baumgartner's published theorem that $\mathrm{MA}_{\aleph_1}$ makes $\omega_1\omega$ a partition ordinal, $\omega_1\omega\to(\omega_1\omega,n)^2$ for every finite $n$ (Baumgartner 1989, §3; the chapter is not held, and its statement is taken from its zbMATH review and from the introduction of Chen, Garti and Weinert). The catalog relation follows by an elementary step: with $n$ the finite Ramsey number for triangles in $k$ colors, a $(k+1)$-coloring of $[\omega_1\omega]^2$ has a color-$0$ homogeneous set of type $\omega_1\omega$ or an $n$-element set colored from the other $k$ colors, hence a monochromatic triangle, and $\omega_1\omega$ is an initial segment of $\omega_1^2$. This bridging deduction is author-recorded here and not independently reviewed. Its only written form found is the color-reduction lemma of an unrefereed, AI-assisted Zenodo deposit (Gao 2026, Lemma 2.1), which reaches the same conclusion from the weaker case $n=3$ that the site quotes; the lemma and the deposit's Theorem 3.1 have their proofs followed on the card, an author-recorded check, and their research reconstructions were independently reviewed and graded faithful and sound. The deposit was the site's one proof claim, listed as partial until its author withdrew it, and it prompted the site's status change; it is not load-bearing for the status, since Baumgartner's theorem in the form above suffices. The deposit's argument is reconstructed step by step, with Baumgartner's relation stated as the imported input in the exact case used, in the research folder for this problem (independently reviewed as it stood on 2026-09-28 and graded faithful and sound in the grade; no tier; the status is unchanged).

ZFC. The cases $k\le2$ are theorems of ZFC: $\omega_1^2\to(\omega_1\omega,3,3)^2$ (Baumgartner and Hajnal 1987, not held, statement from its zbMATH review and from Komjáth 2025, Problem 13 commentary), which contains $k=1$; both cases are on the claimed partial page Baumgartner and Hajnal 1987. Komjáth attributes the case $k=1$, in the form $\omega_1^2\to(\omega_1\alpha,3)^2$ for all $\alpha<\omega_1$, to Erdős and Hajnal [ErHa70]; the attribution is Komjáth's and is not found in that paper, which says in its §1 that it does not investigate relations for order types. The ordinal relations of [ErHa71] are stated under GCH in its review, so $k=1$ has no page of its own and is covered by the 1987 page. Komjáth's Problem 54 discussion records the case $k=3$, $\omega_1^2\to(\omega_1\omega,3,3,3)^2$, as unknown, so the exact question is open in ZFC for $k\ge3$. The triangle targets are sharp: CH gives $\omega_1^2\not\to(\omega_1\omega,4)^2$ (Baumgartner and Hajnal 1987), and $\mathfrak d=\aleph_1$ already suffices (Chen, Garti and Weinert, Theorem 2.9). CH also gives $\omega_1\omega\not\to(\omega_1\omega,3)^2$ (Erdős and Hajnal, as the review of [Ba89b] records), so the route through $\omega_1\omega$ needs a hypothesis beyond ZFC.

Search scope. erdosproblems.com (the problem page, its discussion thread, its proof-claims tab, its revision history and its LaTeX source); the community database (teorth/erdosproblems, entry 1171: status and informal status "not disprovable" since 2026-09-05, formal status unformalized, no formalization); conjectures.io (its results page, its problem catalog and its search: no item for 1171); google-deepmind/formal-conjectures (no ErdosProblems/1171.lean); Zenodo (the deposit's record and its PDF); zbMATH Open (the reviews of [Ba89b] and [BaHa87]); arXiv (a preprint of Golshani, arXiv:2608.13213, on the relation $\omega_1^2\to(\omega_1^2,3)^2$ of Problem 1169, cites this problem and adds nothing to it; a date-sorted query on partition relations for $\omega_1^2$ found nothing else); the Rényi Erdős archive (the scan of [ErHa70]); and the texts of [Ko25], of Chen, Garti and Weinert, and of [Va99]. Not searched: X, MathSciNet, and the Hajnal--Larson Handbook chapter on partition relations. Not held: the PDFs of [Ba89b], [BaHa87] and [ErHa71]; the $k=1$ attribution to [ErHa70] is Komjáth's and is not found in that paper. The bridging step from Baumgartner's theorem is author-recorded here; the only independent review recorded is the 2026-09-28 grade of the research reconstructions noted above, which awards no tier.

Known Results

• Baumgartner 1989 ($\mathrm{MA}_{\aleph_1}$). Under $\mathrm{MA}_{\aleph_1}$, $\omega_1\omega$ and $\omega_1\omega^2$ are partition ordinals: $\omega_1\omega\to(\omega_1\omega,n)^2$ for every finite $n$ (main theorem, §3; statement per the zbMATH review Zbl 0703.03027 and the introduction of Chen, Garti and Weinert; PDF not held). With $n$ the finite Ramsey number for triangles in $k$ colors and $\omega_1\omega$ an initial segment of $\omega_1^2$, this gives $\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$ for every finite $k$ in every model of $\mathrm{MA}_{\aleph_1}$; since $\mathrm{MA}_{\aleph_1}$ is consistent relative to ZFC [SoTe71], the exact catalog statement is not disprovable in ZFC. The bridging step is author-recorded here, not independently reviewed.
• Gao 2026 ($\mathrm{MA}_{\aleph_1}$; unrefereed, AI-assisted). Zenodo deposit of 5 September 2026 (card): Lemma 2.1, if $\alpha\to(\alpha,3)^2$ then $\alpha\to(\alpha,3,\ldots,3)^2_{k+1}$ for every finite $k\ge1$ (merge color $0$ with one triangle color, induct, then split the merged color with the two-color relation on the set of type $\alpha$); Theorem 3.1, $\mathrm{MA}_{\aleph_1}$ implies the exact catalog relation for every $k\ge1$. Was listed on the site as a partial proof claim with no comments until its author withdrew it on 1 October 2026 (see the claim page); it prompted the site's status change of 2026-09-05. Proofs of the lemma and theorem followed on the card (author-recorded); research reconstructions independently reviewed and graded faithful and sound, no tier; not load-bearing given Baumgartner's theorem.
• Baumgartner and Hajnal 1987 (ZFC, $k=2$). $\omega_1^2\to(\omega_1\omega,3,3)^2$ holds in ZFC (positive relation, the case $\kappa=\omega$ of $(\kappa^+)^2\to(\kappa^+\kappa,3,3)^2$ for regular $\kappa$ with $\kappa^{<\kappa}=\kappa$; statement per the zbMATH review Zbl 0635.03042 and Komjáth 2025, Problem 13 commentary); this contains the case $k=1$. Recorded on its claim page as a claimed partial claim. Komjáth attributes the case $k=1$, $\omega_1^2\to(\omega_1\alpha,3)^2$ for $\alpha<\omega_1$, to Erdős and Hajnal [ErHa70]; the attribution is Komjáth's and is not found in that paper.
• Open in ZFC for $k\ge3$. Komjáth 2025 (Problem 54 discussion) records that it is unknown whether $\omega_1^2\to(\omega_1\omega,3,3,3)^2$ holds, or even whether $\alpha\not\to(\beta,4)^2$ together with $\alpha\to(\beta,3,3,3)^2$ is consistent for some ordinals $\alpha,\beta$. The site's summary of Gao's claim likewise says the ZFC case remains open.
• Sharpness of the triangle targets. CH gives $\omega_1^2\not\to(\omega_1\omega,4)^2$ (negative relation, the case $\kappa=\omega$ of $2^\kappa=\kappa^+$ implying $(\kappa^+)^2\not\to(\kappa^+\kappa,4)^2$); $\mathfrak d=\aleph_1$ already suffices (Chen, Garti and Weinert, Theorem 2.9). CH also gives $\omega_1\omega\not\to(\omega_1\omega,3)^2$ (Erdős and Hajnal, as the review of [Ba89b] records), so Baumgartner's route through $\omega_1\omega$ needs a hypothesis beyond ZFC.

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
• baumgartner_1987_remark_partition_relations_infinite_ordinals
• baumgartner_1987_remark_partition_relations_infinite_ordinals / baumgartner_1987_remark_partition_relations_infinite_ordinals
• baumgartner_1987_remark_partition_relations_infinite_ordinals / negative_relation
• baumgartner_1987_remark_partition_relations_infinite_ordinals / positive_relation
• baumgartner_1989_remarks_partition_ordinals
• baumgartner_1989_remarks_partition_ordinals / baumgartner_1989_remarks_partition_ordinals
• baumgartner_1989_remarks_partition_ordinals / main_theorem
• chen_2018_cardinal_characteristics_continuum_partitions
• erdos_1974_unsolved_solved_problems_set_theory
• erdos_1974_unsolved_solved_problems_set_theory / theorem_p273_hajnal
• gao_2026_finite_color_partition_relation_omega_1_squared
• gao_2026_finite_color_partition_relation_omega_1_squared / lemma_2_1
• gao_2026_finite_color_partition_relation_omega_1_squared / theorem_3_1
• komjath_2025_erdos_hajnal_problem_list