---

Claim. Problem 1219 asks whether $\sum_k 2^{\aleph_{n_k}}\to(\aleph_\omega)^2$, with two colors, for an increasing sequence $(n_k)$ of integers such that $2^{\aleph_{n_k}}$ is strictly increasing and $2^{\aleph_{n_0}}>\aleph_\omega$. Shelah's Corollary 1.3 (p. 1260) states: if $\aleph_\omega<2^{\aleph_{n(0)}}<2^{\aleph_{n(1)}}<\cdots$ then

$$\sum_{n<\omega}2^{\aleph_n}\to(\aleph_\omega,\aleph_\omega)^2 ,$$

and the Remark after it records that this answers Problem 3 of the 1971 Erdős--Hajnal list. The two sums are the same cardinal: a countable sum of infinite cardinals is its supremum, $n\mapsto2^{\aleph_n}$ is nondecreasing and $n_k\to\infty$, so $\sum_{n<\omega}2^{\aleph_n}=\sup_k2^{\aleph_{n_k}} =\sum_k2^{\aleph_{n_k}}$, and the relation is the catalog's. The corollary is the case $\lambda=\aleph_\omega$, $\kappa=\omega$ of Theorem 1.2 (p. 1260): if $\kappa\to(\kappa)^2_2$, $\kappa=\operatorname{cf}\lambda$, and $\langle2^\mu:\mu<\lambda\rangle$ is not eventually constant but eventually at least $\lambda$, then $\sum_{\mu<\lambda}2^\mu\to(\lambda)^2_2$, indeed $\to(\lambda,\lambda,\omega)^2$; here $\kappa\to(\kappa)^2_2$ is Ramsey's theorem. The proof takes half a page from the paper's Canonization Lemma 1.1 (pp. 1258--1260), which reduces a two-coloring of pairs on a union of blocks $A_i$ of size $(2^{\mu(i)})^+$ to a coloring of pairs of indices, to which Ramsey's theorem is applied. Section 0 (p. 1257) describes Problem 3 as the only open case, for infinite cardinals, of $\lambda\to(\mu)^2_2$, so the corollary completes that discussion; this answers the problem in the affirmative.

Misprint in Theorem 1.2. The printed hypothesis says the powers are eventually at least $\kappa$, a misprint for $\lambda$: the proof chooses $\mu(i)$ with $2^{\mu(i)}\ge\lambda$, and as printed the theorem would assert $\aleph_\omega\to(\aleph_\omega)^2_2$ whenever $2^{\aleph_n}=\aleph_{n+1}$ for all $n$, which fails for every singular cardinal. Corollary 1.3 carries the catalog's hypothesis $2^{\aleph_{n(0)}}>\aleph_\omega$ explicitly, so the claim does not depend on this reading. Hajnal's stronger conjecture $\sum_k2^{\aleph_{n_k}}\to(\aleph_\omega,4)^3$ (Conjecture 1A, p. 1261) is a separate question, recorded as still unproven by Komjáth (2025), and is not part of this claim.

Source. Saharon Shelah, Notes on partition calculus, in Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III, Colloq. Math. Soc. János Bolyai 10, North-Holland, Amsterdam, 1975, pp. 1257--1276; MR 0406798; zbMATH 0325.04005; Shelah archive Sh:40, whose copy of the printed article is the second link above. The source card carries result pages for Corollary 1.3 and Theorem 1.2. An author-recorded reconstruction of the proofs, with the identification of the two sums written out, is filed in the research folder for this problem; it is not an independent review. The volume carries only the year, so this page is dated the first of January 1975.

Acceptance. Reviewed: the curator of erdosproblems.com, T. F. Bloom, marks the problem PROVED and his remark credits the proof to Shelah [Sh75] (problem page last edited 1 September 2026; as of 2026-10-07 no comments and no proof claims); Komjáth's survey (Bull. Symbolic Logic 31, 2025, p. 419) records the paper as the proof of Problem 3 and the last remaining case of the discussion of $\lambda\to(\kappa)^2_2$, a named expert's documented acceptance; the community database marks the problem proved, in an entry last updated 2026-09-12. The curator and Komjáth are independent of the author. The paper appeared in a colloquium proceedings volume, Colloq. Math. Soc. János Bolyai 10, reviewed by Mathematical Reviews and zbMATH, not in a journal, so refereed is not listed. Nothing on this page is independently reviewed by this project.

Depends on.

• Shelah, Corollary 1.3
• Shelah, Theorem 1.2