Erdős, P. and Hajnal, A., Unsolved problems in set theory. Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) (1971), 17-48. Not held; the question is Problem 3 of this list, as [Sh75] (p. 1257 and the Remark on p. 1260) and [Ko25b] (p. 419) both record.

P. Komjáth, The Erdős--Hajnal Problem List. Bull. Symb. Log. 31 (2025), 418--461, doi:10.1017/bsl.2025.1 (the site's bibliography prints "Probem"). Problem 3 (Erdős, Hajnal, Rado), p. 419, states the relation as $\lambda\to(\aleph_\omega)^2_2$ for $\lambda=2^{\aleph_{n_0}}+2^{\aleph_{n_1}}+\cdots$ under $\aleph_\omega<2^{\aleph_{n_0}}<2^{\aleph_{n_1}}<\cdots$, records that Shelah proved it in [152] as "the last remaining case of the discussion of the relation $\lambda\to(\kappa)^2_2$", and records Hajnal's conjecture $\lambda\to(\aleph_\omega,4)^3$ as so far unproven. Library home: Komjáth 2025.

Shelah, S., Notes on partition calculus. 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), 1257--1276; MR 0406798, zbMATH 0325.04005, Shelah archive Sh:40. The Shelah archive's copy of the printed article is at https://shelah.logic.at/files/95045/40.pdf; §0, p. 1257, names Problem 3 of [ErHa71] as "the only open case (for infinite cardinals) of $\lambda\to(\mu)^2_2$" and solves it affirmatively; Theorem 1.2, p. 1260, with its half-page proof from the Canonization Lemma 1.1, pp. 1258--1260; Corollary 1.3, p. 1260, "If $\aleph_\omega<2^{\aleph_{n(0)}}<2^{\aleph_{n(1)}}<\cdots$ then $\sum_{n<\omega}2^{\aleph_n}\to(\aleph_\omega,\aleph_\omega)^2$", with the Remark that this answers Problem 3; Conjecture 1A (Hajnal), p. 1261. Library home: Shelah 1975 and its Shelah 1975, Corollary 1.3 page.