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Statement

Notation, standard and used by the paper without restatement for cardinals: $\lambda\to(\mu_0,\mu_1)^3$ holds when every two-coloring of the triples from a set of size $\lambda$ has a subset of size $\mu_0$ all of whose triples have the first color or a subset of size $\mu_1$ all of whose triples have the second, and $\not\to$ denotes its failure.

Conjecture 1A (Hajnal; printed p. 1261). Let $\aleph_\omega<2^{\aleph_{n(0)}}<2^{\aleph_{n(1)}}<\cdots$. Then $\sum_{n<\omega}2^{\aleph_n}\to(\aleph_\omega,4)^3$.

The hypothesis is that of Corollary 1.3, and the conjecture strengthens the corollary's two-dimensional relation to triples. The Remark following it (p. 1261) gives its motivation and its limit: wherever $\lambda\to(\mu,\mu)^2$ had been proved before this paper, $\lambda\to(\mu,4)^3$ holds as well, whereas $\sum_{n<\omega}2^{\aleph_n}\not\to(\aleph_\omega,5)^3$. Neither statement in the Remark is proved on the page.

Standing. A conjecture as posed, not a result of the paper. Komjáth's 2025 survey, p. 419 (komjath_2025_erdos_hajnal_problem_list), records that Hajnal conjectured $\lambda\to(\aleph_\omega,4)^3$ under the same assumption and that the conjecture "has so far been unproven". No later resolution was found in the search recorded on Problem 1219's page.

Source. Saharon Shelah, Notes on partition calculus, Infinite and finite sets (Keszthely, 1973), Colloq. Math. Soc. János Bolyai 10, North-Holland, 1975, 1257--1276; Conjecture 1A and its Remark on printed p. 1261 (PDF p. 5 of the archive's scan), read on the page image. The artifact is identified in the source digest.

Read depth. Claims checked: the conjecture and the Remark were read clause by clause on the page image. Nothing here is independently reviewed.

Proof pointer

None; the statement is a conjecture. The negative relation $\not\to(\aleph_\omega,5)^3$ in the Remark is asserted without proof or citation on p. 1261.

Dependencies

None consumed. The conjecture presupposes the hypothesis and the notation of Corollary 1.3.

Bears on

• Problem 1219: the stronger three-dimensional question the site records as open after Komjáth; it is not part of the catalog question, whose two-dimensional relation Corollary 1.3 proves.