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Source. Richard Laver, An $(\aleph_2,\aleph_2,\aleph_0)$-saturated ideal on $\omega_1$ (1982), § 3: the statement on p. 177, its announcement on p. 173, and the sketch of proof on pp. 177--178. The edition is identified on the source card.

Statement. Assume $2^{\aleph_0}=\aleph_1$ and that there is a normal $(\aleph_2,\aleph_2,\aleph_0)$-saturated $\omega_1$-ideal, in the sense defined on the main theorem's page. Then $\omega_2\to(\omega_1+\omega_1+1,\alpha)^2$ for every $\alpha<\omega_2$: for every partition $[\omega_2]^2=R\cup G$ there is a set of order type $\omega_1+\omega_1+1$ all of whose pairs lie in $R$, or a set of order type $\alpha$ all of whose pairs lie in $G$.

The introduction's announcement (p. 173) omits the word normal; the statement of § 3 (p. 177) has it, and the ideal built in § 1 is normal. The paper writes $\omega_1+\omega_1+1$; Komjáth's survey and Problem 1172's page write $\omega_1 2+1$, the same ordinal.

Consequences. With $\alpha=\omega_1+\omega$ the theorem gives $\omega_2\to(\omega_1+\omega)^2_2$, since a set of order type $\omega_1+\omega_1+1$ contains one of order type $\omega_1+\omega$; with $\alpha=\omega_1+\omega_1+1$ it gives $\omega_2\to(\alpha)^2_2$ for every $\alpha\le\omega_1+\omega_1+1$. The model of § 1 satisfies $2^{\aleph_0}=\aleph_1$ (p. 173) and has a normal such ideal, so both relations hold there, relative to the consistency of a huge cardinal. The introduction places § 3 after recalling the question of Erdős and Hajnal (the paper's [3], their 1971 problem list) whether $\omega_2\to(\omega_1+\omega)^2_2$ is consistent (p. 173), but the paper does not state these consequences in words; they are one-line deductions made here.

Proof pointer. Pages 177--178; the paper says "We just give a sketch modulo familiar arguments". Fix $[\omega_2]^2=R\cup G$, and for a set $B\subseteq\omega_2$ of order type $\omega_1$ copy the ideal to $B$ through the order isomorphism with $\omega_1$. The cases are whether there are $\aleph_2$ sets $B_\alpha$ of order type $\omega_1$, placed one above another, with all pairs inside each $B_\alpha$ in $R$, and such that for $\alpha<\beta$ a set in the dual filter on $B_\alpha$ has all its pairs with $B_\beta$ in $G$. If there are, a ramification argument builds, for each $\alpha<\omega_2$, a set of order type $\alpha$ with all pairs in $G$. If not, the failed step of the construction gives a set $B_\alpha$ of order type $\omega_1$ with all pairs in $R$ and $\aleph_2$ points each joined in $R$ to a positive part of $B_\alpha$; $(\aleph_2,\aleph_2,\aleph_0)$-saturation makes the countable intersections of these parts positive, and a ramification argument extends $B_\alpha$ to a set of order type $\omega_1+\omega_1+1$ with all pairs in $R$. Building the $B_\alpha$ uses a ramification argument in the manner of Kurepa and of Erdős and Rado, and a generalization of an argument of Galvin (credited to the paper's [1], Baumgartner, Hajnal and Máté) that under $2^{\aleph_0}=\aleph_1$, among any $\aleph_2$ sets in a normal filter on $\omega_1$ there are $\aleph_1$ whose intersection is in the filter. The paper adds that a forthcoming paper of Kanamori will give a similar proof that $\kappa^+\to(\kappa+\kappa+1,(\kappa:\kappa^+))^2$ for weakly compact $\kappa$. Not reconstructed here.

Read depth. Claims checked: the statement on p. 177 and its announcement on p. 173 were read clause by clause on the page images. The sketch on pp. 177--178 was read on the page images and no step was checked; the sketch itself leaves its ramification arguments to the reader. Nothing here is independently reviewed.

Dependencies. For the consequences in a model: the main theorem. External: the Galvin argument as given by Baumgartner, Hajnal and Máté, Weak saturation properties of ideals (Colloq. Math. Soc. János Bolyai 10, 1975, 137--158, the paper's [1]), and the ramification method of Kurepa (1952, [9]) and of Erdős and Rado (A partition calculus in set theory, 1956, [2]). The first two have no card in the library.

Bears on.

• #1172: in the model of § 1 the theorem gives $\omega_2\to(\omega_1+\omega)^2_2$ relative to a huge cardinal; that model satisfies $2^{\aleph_0}=\aleph_1$ and $2^{\aleph_1}=\aleph_2$, and the paper does not state that it satisfies GCH, which the problem's consistency question requires (see the main theorem's page).
• #1170: in the same model it gives $\omega_2\to(\alpha)^2_2$ for every $\alpha\le\omega_1+\omega_1+1$; the problem asks for every $\alpha<\omega_2$, which the paper does not address.