Laver: An (ℵ₂, ℵ₂, ℵ₀)-saturated ideal on ω₁

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main_theorem: Laver's § 1 theorem that the consistency of a huge cardinal gives the consistency of a normal (ℵ₂, ℵ₂, ℵ₀)-saturated ideal on ω₁, in a model of 2^ℵ₀ = ℵ₁ and 2^ℵ₁ = ℵ₂; the paper does not state that GCH holds there.

theorem_p177: Laver's § 3 theorem, proved in sketch, that 2^ℵ₀ = ℵ₁ and a normal (ℵ₂, ℵ₂, ℵ₀)-saturated ideal on ω₁ give ω₂ → (ω₁ + ω₁ + 1, α)^2 for every α < ω₂, so the model of § 1 satisfies ω₂ → (ω₁ + ω)^2_2.

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Richard Laver, An $(\aleph_2,\aleph_2,\aleph_0)$-saturated ideal on $\omega_1$, in D. van Dalen, D. Lascar and J. Smiley (eds.), Logic Colloquium '80, Studies in Logic and the Foundations of Mathematics 108, North-Holland, 1982, pp. 173--180, doi:10.1016/S0049-237X(09)70510-5; the author at the University of Colorado, Boulder. Cited as [La82] on Problem 1170's page and as [120] in Komjáth's survey of the Erdős–Hajnal problem list.

The copy read for this card is the publisher's PDF from ScienceDirect: 8 pages, printed pp. 173--180 = PDF pp. 1--8 (printed p. $n$ is PDF p. $n-172$), a scan of the printed chapter with an OCR text layer that garbles most of the mathematics, so the statements below were read on rendered page images. The file prints "© North-Holland Publishing Company, 1982" in the header of its first page (p. 173). The Crossref record (https://api.crossref.org/works/10.1016/S0049-237X(09)70510-5, read 2026-10-14) names Elsevier as publisher and lists only text-and-data-mining licenses, with no Creative Commons license: every other right reserved.

What the paper proves. Kunen had obtained an $\aleph_2$-saturated ideal on $\omega_1$ from a huge cardinal, and Magidor from an almost huge one (p. 173). § 1 modifies Kunen's construction to obtain, from a huge cardinal, a normal $(\aleph_2,\aleph_2,\aleph_0)$-saturated ideal on $\omega_1$: every $\aleph_2$ positive sets include $\aleph_2$ of which every countably many have positive intersection (main theorem). § 2 shows that $2^{\aleph_0}=\aleph_1$ and an $(\aleph_2,\aleph_2,\aleph_0)$-saturated $\omega_1$-ideal give the polarized relation $\binom{\aleph_2}{\aleph_1}\to\binom{\aleph_1}{\aleph_1}^{1,1}_{\aleph_0}$: for every $f:\omega_2\times\omega_1\to\omega$ there are sets $A\subseteq\omega_2$ and $B\subseteq\omega_1$ of size $\aleph_1$ with $f$ constant on $A\times B$ (p. 176). The paper recalls that its [3], the Erdős–Hajnal problem list of 1971, asked whether the weaker $\binom{\aleph_2}{\aleph_1}\to\binom{\aleph_0}{\aleph_1}^{1,1}_2$ is consistent with $2^{\aleph_0}=\aleph_1$, and it reports Galvin's extension of § 2 to $\binom{\omega_2}{\omega_1}\to\binom{\alpha}{\omega_1}^{1,1}_{\aleph_0}$ for every $\alpha<\omega_2$. § 3 proves, in sketch, that $2^{\aleph_0}=\aleph_1$ and a normal such ideal give $\omega_2\to(\omega_1+\omega_1+1,\alpha)^2$ for every $\alpha<\omega_2$ (theorem of p. 177), after recalling that [3] asked whether $\omega_2\to(\omega_1+\omega)^2_2$ is consistent. § 4 strengthens the saturation in the model of § 1, adding a normal ideal on $\omega_2$ with a countably closed dense set, and shows that the variant model in which $\kappa$ becomes $\aleph_2$ and $\lambda$ becomes $\aleph_3$ satisfies the Chang-type relation $\aleph_3\to(\aleph_2)^{\aleph_0}_{\aleph_2/\aleph_1}$. By p. 173, every result except that last one has an analogue with $\omega_1$ replaced by the successor of an arbitrary regular cardinal.

GCH. The paper does not state that GCH holds in its model. The model of § 1 satisfies the proposition $\Phi$, whose first clauses are $2^{\aleph_0}=\aleph_1$ and $2^{\aleph_1}=\aleph_2$ (p. 174, with pp. 173 and 175); the paper says nothing about $2^{\aleph_2}$ or larger powers, or about the ground model's continuum function, and mentions GCH only on p. 177, as the hypothesis of a question about the polarized relations of § 2. The main theorem's page gives the passages. Komjáth's survey reports the result as the consistency of GCH with such an ideal (Problem 24, p. 431, of the survey); that is the survey's wording, not the paper's. The ideal of § 1 is normal (pp. 173 and 175).

Read status: claims checked. The definitions and announcements of p. 173, the remark on Woodin's dense ideal on p. 174, the proposition $\Phi$ with Kunen's clauses (a)--(c) on p. 174, the assertions of p. 175 that the new iteration has the $(\kappa,\kappa,<\beta)$-cc, that $2^\kappa=\lambda$ in the final model and that the ideal is normal and nontrivial, the § 2 statement and Galvin's extension on p. 176, the GCH question, the remark on consistency relative to ZFC and the § 3 statement on p. 177, the § 4 statements on pp. 178--179, and the reference list (pp. 179--180) were read clause by clause on the page images. The proofs of §§ 1--4 were looked through on the page images and no proof step was checked; the proof of § 3 is itself a sketch. Nothing here is independently reviewed.

Bears on. #1172: the model of § 1 satisfies $2^{\aleph_0}=\aleph_1$ and has a normal $(\aleph_2,\aleph_2,\aleph_0)$-saturated ideal on $\omega_1$, so by the § 3 theorem it satisfies $\omega_2\to(\omega_1+\omega)^2_2$, relative to a huge cardinal (a one-line deduction made here). The ideal is normal, but the paper states only $2^{\aleph_0}=\aleph_1$ and $2^{\aleph_1}=\aleph_2$ in that model, not GCH, which the problem's consistency question requires. #1170: in the same model $\omega_2\to(\alpha)^2_2$ holds for every $\alpha\le\omega_1+\omega_1+1$, a partial instance of the problem; the paper says nothing about larger $\alpha<\omega_2$.

Results.

• Main theorem (§ 1, pp. 173--176): from a huge cardinal, a normal $(\aleph_2,\aleph_2,\aleph_0)$-saturated ideal on $\omega_1$, with what the paper says about the continuum function of its model.
• Theorem of p. 177 (§ 3, pp. 177--178): $\omega_2\to(\omega_1+\omega_1+1,\alpha)^2$ for every $\alpha<\omega_2$, from $2^{\aleph_0}=\aleph_1$ and a normal such ideal.

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