Problem 1171

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

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

Source: erdosproblems.com/1171

Set theory
Ramsey theory

Status

Open

No claim settles this problem.

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and Hajnal 1987; the deposit that prompted the site's label has its own page, Gao's conditional proof, recorded as withdrawn.

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.
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 and Hajnal 1987.
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 Unrefereed, AI-assisted; withdrawn by its author on 1 October 2026 (the Zenodo record is a tombstone giving retraction or withdrawal as the reason).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.
Ko25: Komjáth, Péter, The Erdős–Hajnal problem list. Bull. Symbolic Logic 31 (2025), 418-461. DOI 10.1017/bsl.2025.1. Library home: Komjáth 2025.
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.

Problem 1171

Status

Proof claims

Discussion