Problem 1172

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claims/: The 1 claim page of Problem 1172, one per claimant's result; the problem's standing derives from them.

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Statement. Establish whether the following are true assuming the generalised continuum hypothesis:

$$\omega_3 \to (\omega_2,\omega_1+2)^2,$$ $$\omega_3\to (\omega_2+\omega_1,\omega_2+\omega)^2,$$ $$\omega_2\to (\omega_1^{\omega+2}+2, \omega_1+2)^2.$$

Establish whether the following is consistent with the generalised continuum hypothesis:

$$\omega_2\to (\omega_1+\omega)_2^2,$$

or even $\omega_2 \to (\xi)_2^2$ for all $\xi<\omega_2$.

Formulation. The three relations asked under GCH come from the booklet item [Va99, 7.87]. Its public scan is cut off at the page edge after "$\omega_3\to(\omega_2$", and the left side of the final relation is cut off too. An earlier version of the site page said that the right-hand sides of the first and final statements were missing from the booklet and might have been filled in incorrectly. The final statement was later taken from [ErHa74, p. 272]. None of the site's sources prints the first relation in full. As printed, the first relation follows from the Erdős–Rado theorem that the site's remark quotes. Under GCH, $(2^{\aleph_1})^+=\omega_3\to(\omega_2+1)^2_{\aleph_1}$, so every $2$-coloring of $[\omega_3]^2$ has a homogeneous set of type $\omega_2+1$, which contains sets of types $\omega_2$ and $\omega_1+2$. The standing answers the site's wording. Its first relation holds; the other two relations and the consistency question are open.

Status. Open. The site's label is OPEN.

Source. erdosproblems.com/1172, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1172, https://www.erdosproblems.com/1172.

References.

• [ErRa56] Erdős, P. and Rado, R., A partition calculus in set theory. Bull. Amer. Math. Soc. (1956), 427-489.
• [ErHa74] Erdős, P. and Hajnal, A., Unsolved and solved problems in set theory. Proc. Sympos. Pure Math. 25 (1974), 269-287.
• [Va99] Some of Paul's favorite problems, booklet for the conference "Paul Erdős and his mathematics", Budapest, July 1999; item 7.87. Library home: various_1999_some_pauls_favorite_problems.

Formalization. None recorded.

Current assessment

The standing judges the statement above, as the site gave it on 2026-09-04 (page last edited 11 April 2026). Its first relation, as printed, follows from the Erdős–Rado theorem under GCH; the step is recorded on the Erdős–Rado page, an accepted partial claim. The second relation, $\omega_3\to(\omega_2+\omega_1,\omega_2+\omega)^2$, the third relation, $\omega_2\to(\omega_1^{\omega+2}+2,\omega_1+2)^2$, and the consistency of $\omega_2\to(\omega_1+\omega)^2_2$ with GCH are open, so the problem is open. Komjáth's Problem 10/B (source card) records that Erdős and Hajnal proved $\omega_2\to(\omega_1+n)^2_2$ under CH for finite $n$, and that the consistency question stays posed. Baumgartner, Hajnal and Todorčević, Extensions of the Erdős–Rado theorem (1993, Zbl 0846.03021), prove under CH at $\kappa=\omega_1$ a relation $\omega_2\to(\omega_1^{\omega+2}+1,(\omega_1+n)_k)^2$ close to the third relation, which does not settle it. No literature search beyond these sources and the site is recorded.

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.

• erdos_1956_partition_calculus_set_theory
• erdos_1974_unsolved_solved_problems_set_theory
• erdos_1974_unsolved_solved_problems_set_theory / question_p272
• komjath_2025_erdos_hajnal_problem_list