Problem 1167

../

claims/: The 2 claim pages of Problem 1167, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $r\geq 2$ be finite and $\lambda$ be an infinite cardinal. Let $\kappa_\alpha$ be cardinals for all $\alpha<\gamma$.

Is it true that

$$2^\lambda \to (\kappa_\alpha+1)_{\alpha<\gamma}^{r+1}$$

implies

$$\lambda \to (\kappa_\alpha)_{\alpha<\gamma}^{r}?$$

Here $+$ means cardinal addition, so that $\kappa_\alpha+1=\kappa_\alpha$ if $\kappa_\alpha$ is infinite.

Statement (corrected). Let $r\geq 2$ be finite and $\lambda$ be an infinite cardinal. Let $\gamma\geq 2$ and let $\kappa_\alpha>r$ be cardinals for all $\alpha<\gamma$.

Is it true that

$$2^\lambda \to (\kappa_\alpha+1)_{\alpha<\gamma}^{r+1}$$

implies

$$\lambda \to (\kappa_\alpha)_{\alpha<\gamma}^{r}?$$

Here $+$ means cardinal addition, so that $\kappa_\alpha+1=\kappa_\alpha$ if $\kappa_\alpha$ is infinite.

Notes. The site's wording follows the booklet item [Va99, 7.79] and puts no condition on $\gamma$ or on the $\kappa_\alpha$, and, read as the site words it, the implication is false. For $\gamma=1$ and $\kappa_0=\lambda^+$, the premise $2^\lambda\to(\lambda^+)^{r+1}_1$ says only that $2^\lambda\ge\lambda^+$, which holds, while $\lambda\to(\lambda^+)^r_1$ needs a subset of $\lambda$ of size $\lambda^+$. For $\gamma=2$, $\kappa_0=\lambda^+$ and $\kappa_1=r$, the premise holds: either some $(r+1)$-set has color $1$, or all of $2^\lambda$ is homogeneous in color $0$. The constant coloring of $[\lambda]^r$ with color $0$ refutes the conclusion. Both failures are boundary cases of a dropped range, and the change adds the conditions $\gamma\ge2$ and $\kappa_\alpha>r$ of the Erdős–Hajnal list, which exclude them. Komjáth's Problem 2 ([Ko25b], p. 419) states the question for finite $r$ with $\kappa_\nu>r$ and a condition on $\gamma$ that the site's curator reads as $\gamma\ge2$, taking the printed inequality for a misprint. The curator's reply in the discussion thread points to these conditions rather than accepting a disproof, and the site keeps the label OPEN. The formal-conjectures statement file adds $\gamma\ge2$ and $\kappa_\alpha>r$ and proves the first counterexample as its test lemma erdos_1167.unrestricted_is_false. That lemma is in a statement file, so it is not a formalization link and gets no claim page, and the corpus has not built it. Rafik Zeraoulia gave the first counterexample in a note of 31 January 2026. It answers the site's wording, not the corrected Statement, so it does not count toward the problem's standing; it is credited here and recorded on Zeraoulia's rejected claim page. The problem's standing judges the corrected Statement.

Status. The site's label is OPEN. The corrected Statement, with the conditions $\gamma\ge2$ and $\kappa_\alpha>r$, is open; the counterexample to the site's wording at $\gamma=1$ is credited in the Notes and recorded on a rejected claim page.

Source. erdosproblems.com/1167, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1167, https://www.erdosproblems.com/1167.

References.

• [ErHa71] Erdős, P. and Hajnal, A., Unsolved problems in set theory. Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I) (1971), 17-48.
• [EHMR84] Erdős, P., Hajnal, A., Máté, A. and Rado, R., Combinatorial set theory: partition relations for cardinals. Studies in Logic and the Foundations of Mathematics 106, North-Holland (1984).
• [Va99] Some of Paul's favorite problems, booklet for the conference "Paul Erdős and his mathematics", Budapest, July 1999; item 7.79.
• [Ko25b] P. Komjáth, The Erdős-Hajnal Problem List. Bull. Symb. Log. (2025), 418-461; Problem 2, p. 419 (source card).

Formalization. Statement in formal-conjectures.

Current assessment

The site's Statement, as the site gave it on 2026-09-04 (page last edited 1 September 2026), puts no condition on $\gamma$ or on the $\kappa_\alpha$, and, read as the site words it, it is false: for $\gamma=1$ and $\kappa_0=\lambda^+$ the premise holds and the conclusion fails. Rafik Zeraoulia gave this counterexample in a note of 31 January 2026, recorded as rejected on Zeraoulia's page, since it answers the site's wording, not the corrected Statement. The site labels the problem OPEN, and its curator's reply in the discussion thread points to the conditions of the Erdős–Hajnal list rather than accepting a disproof. The corrected Statement, which adds those conditions, is the statement this page's standing judges.

The corrected Statement, with the list's conditions $2\le r<\omega$, $\gamma\ge2$ and $\kappa_\alpha>r$, is open. Erdős, Hajnal, Máté and Rado prove five cases of it (see Known Results), and the case Erdős and Hajnal named as the most difficult in [ErHa71], $r=2$ with one singular $\kappa_\alpha$ and the others finite, is open even under GCH. This page records no current literature search beyond the site, its thread, the formal-conjectures file and Komjáth's survey.

Known Results

Erdős, Hajnal, Máté and Rado [EHMR84] prove the implication, under the list's conditions, in five cases: all $\kappa_\alpha$ finite; $\kappa_0$ and $\kappa_1$ infinite with $\kappa_0$ regular; $r\ge3$ with $\kappa_0$ infinite and regular; $r\ge3$ with $\kappa_0$ and $\kappa_1$ infinite; $r\ge4$ with $\kappa_0$ infinite. Komjáth's Problem 2 commentary records the same cases from their Section 24. These are partial results on the corrected Statement, recorded on [[problems/set_theory/E1167/claims/1984_01_01_erdos_hajnal_mate_rado|the monograph's page]].