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Statement

Notation, standard and used by the paper without restatement for cardinals: $\lambda\to(\mu)^2_2$ means that every two-coloring of the pairs from a set of size $\lambda$ has a subset of size $\mu$ all of whose pairs have one color, and $\lambda\to(\mu_0,\mu_1,\mu_2)^2$ that every three-coloring has a subset of size $\mu_i$ homogeneous in color $i$ for some $i$.

Theorem 1.2 (printed p. 1260). Let $\lambda$ be an infinite cardinal with $\kappa=\operatorname{cf}\lambda$, let $\kappa\to(\kappa)^2_2$, and suppose that the sequence $\langle 2^\mu:\mu<\lambda\rangle$ is not eventually constant but is eventually $\ge\lambda$. Then $\chi=\sum_{\mu<\lambda}2^\mu\to(\lambda)^2_2$, and in fact $\chi\to(\lambda,\lambda,\omega)^2$.

Reading note. The printed hypothesis reads "is not eventually constant, but is eventually $\ge\kappa$". That bound is a misprint for $\ge\lambda$: the proof chooses $\mu(i)<\lambda$ with $2^{\mu(i)}\ge\lambda$, which needs the powers eventually at least $\lambda$, and as printed the theorem would apply, with $\kappa=\omega$ and $\lambda=\aleph_\omega$, whenever $2^{\aleph_n}=\aleph_{n+1}$ for all $n$, asserting $\aleph_\omega\to(\aleph_\omega)^2_2$, which fails for every singular cardinal (color a pair by whether its two points lie in the same piece of a partition of $\lambda$ into $\operatorname{cf}\lambda$ pieces of size below $\lambda$). The statement above carries the corrected bound; Corollary 1.3 states its instance, $2^{\aleph_{n(0)}}>\aleph_\omega$, explicitly. The printed proof treats a two-coloring, and the three-color form in the parenthesis is stated without a separate argument.

Source. Saharon Shelah, Notes on partition calculus, Infinite and finite sets (Keszthely, 1973), Colloq. Math. Soc. János Bolyai 10, North-Holland, 1975, 1257--1276; Theorem 1.2 with its proof on printed p. 1260 (PDF p. 4 of the archive's scan), the Canonization Lemma 1.1 on pp. 1258--1260 (PDF pp. 2--4), read on the page images. The artifact is identified in the source digest.

Read depth. Claims checked: the statement was read clause by clause on the page image on 2026-09-27, together with the hypothesis actually used in the proof. The proof (half a page, p. 1260) and the statement and proof of Lemma 1.1 (pp. 1258--1260) were read on the page images for structure only; no step was checked, and the cited relations from [4] were not consulted. Nothing here is independently reviewed.

Proof pointer

Page 1260, in outline. Let $f$ two-color the pairs of $\chi$. Choose cardinals $\mu(i)<\lambda$ for $i<\kappa$ with $\sum_{i<\kappa}\mu(i)=\lambda$, with $2^{\mu(i)}$ strictly increasing in $i$ and with $2^{\mu(i)}\ge\lambda$; put $\lambda_i=(2^{\mu(i)})^+$ and split $\chi$ into consecutive blocks $A_i$ of size $\lambda_i$. If some block contains a set of size at least $\lambda$ on which $f$ is constant, the theorem holds. Otherwise the relations $\lambda_i\to(\lambda_i,\mu(i))^2$ and $\lambda_i\to(\mu(i),\lambda_i)^2$, cited from [4], give inside every subset of $A_i$ of full size $\lambda_i$ sets $B_{i,0}$ and $B_{i,1}$ of size $\mu(i)$ on which $f$ is constantly $0$ and constantly $1$. This realizability inside every full-size subset is the property $P_\alpha$ that the Canonization Lemma 1.1 requires, so the lemma yields $B_\alpha=B_{\alpha,0}\cup B_{\alpha,1}\subseteq A_\alpha$ of size $\mu(\alpha)$ such that, by its clause (1B), the value $f(a,b)$ for $a\in B_i$, $b\in B_j$, $i<j$, depends only on $(i,j)$; call it $g(i,j)$. Since $\kappa\to(\kappa)^2_2$, there are $I\subseteq\kappa$ of size $\kappa$ and a color $\delta$ with $g$ constantly $\delta$ on the pairs from $I$. Then $B=\bigcup_{\alpha\in I}B_{\alpha,\delta}$ has size $\sum_{\alpha\in I}\mu(\alpha)=\lambda$, and $f$ is constantly $\delta$ on its pairs: within one $B_{\alpha,\delta}$ by its homogeneity, across blocks by $g$. Not reconstructed here: the choice of the $\mu(i)$ from the hypothesis, the hypotheses of Lemma 1.1 for these $\lambda_i$ (the paper's growth condition $\prod_{i<j}\lambda_i^{\mu(i)}<\lambda_j$ and $2^{\chi+\kappa}< \lambda_0$ for its $\chi=2$), and the three-color form. Those steps, the proof of the Canonization Lemma 1.1, and a derivation of the three-color form from the two-color one are written out, author-recorded, in the reconstruction of Theorem 1.2 and the reconstruction of Lemma 1.1; those pages are not an independent review and change no standing here.

Dependencies

Within the paper: the Canonization Lemma 1.1 (p. 1258, proof pp. 1258--1260), whose clause (1B) supplies the reduction to a coloring of block indices. Outside it: the relations $\lambda_i\to(\lambda_i,\mu(i))^2$ and $\lambda_i\to(\mu(i),\lambda_i)^2$ for $\lambda_i=(2^{\mu(i)})^+$, cited to [4] (Erdős, Hajnal and Rado, Partition relations for cardinals, Acta Math. Acad. Sci. Hungar. 16 (1965), 93--196, not held), and the hypothesis $\kappa\to(\kappa)^2_2$, which for $\kappa=\omega$ is Ramsey's theorem.

Bears on

• Problem 1219: through its case $\lambda=\aleph_\omega$, $\kappa=\omega$, which is Corollary 1.3, the problem's relation; the Remark after the corollary records that, with this theorem, the question of which infinite $\lambda,\mu$ satisfy $\lambda\to(\mu)^2_2$ is fully answered.