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Statement

Conclusion 3.5 (p. 376). In ZFC:

1. If $n_i<\omega$ for $i<\omega$ and, for every $i<\omega$, there is $m$ such that $\aleph_j\not\to[\aleph_{n_i}]^{<\omega}_{\aleph_i}$ for all $j>m$, then $\aleph_{\omega+1}\not\to[\aleph_{\omega+1}]^2_{\aleph_{\omega+1}}$. The paper introduces the clause as an example ("E.g.").
2. $\lambda^+\not\to[\lambda^+]^2_{\aleph_0}$.
3. If $\lambda$ is inaccessible and not Mahlo, then $\lambda\not\to[\lambda]^2_{\aleph_0}$.
4. $\aleph_{\omega_1}^+\not\to[\aleph_{\omega_1}^+]^2_{\aleph_1}$.

Part (2) carries no printed hypothesis on $\lambda$; its proof applies Theorem 3.3(1) to the set of limit ordinals between $\lambda$ and $\lambda^+$.

The introduction's form (B) (p. 356). If for every $n<\omega$ there are $m,k$ with $\aleph_{m'}\not\to[\aleph_k]^{<\omega}_{\aleph_n}$ for all $m'>m$, which the introduction glosses as "various instances of the Chang conjecture fail", then $\aleph_{\omega+1}\not\to[\aleph_{\omega+1}]^2_{\aleph_{\omega+1}}$. The introduction also records Todorcevic's $\lambda^+\not\to[\lambda^+]^2_{\operatorname{cf}\lambda}$ when $\mu^{\operatorname{cf}\lambda}<\lambda$ for all $\mu<\lambda$.

Source. Saharon Shelah, Was Sierpiński right? I, Israel J. Math. 62 (1988), no. 3, 355--380, doi:10.1007/BF02783304: Conclusion 3.5 and its proof on p. 376, Theorem 3.3 on pp. 371--372 with its proof on pp. 372--375, the introduction's (B) on p. 356. The edition is identified on the source card.

Read depth. Claims checked: the four statements and the introduction's (B) were read clause by clause on the printed page, and the one-line derivations of each part from Theorem 3.3 were read. The proof of Theorem 3.3 (pp. 372--375) was not checked.

Proof pointer

Page 376, each part from Theorem 3.3 (pp. 371--372), which colors pairs from a regular $\lambda$ using stationary sets $S_i$ of points of fixed cofinality $\theta_i$ that reflect in no inaccessible, together with colorings $g_\kappa$ of finite subsets of each regular $\kappa<\lambda$. Part (1): Theorem 3.3(2) gives $\aleph_{\omega+1}\not\to[\aleph_{\omega+1}]^2_{\aleph_\omega}$, with $g_m$ on the finite subsets of $\aleph_m$ chosen from the failures of the Chang-type relations, and Theorem 3.3(3) gives the stronger version with $\aleph_{\omega+1}$ colors. Part (2): Theorem 3.3(1) applied to $S=\{\delta<\lambda^+:\delta\text{ a limit}>\lambda\}$. Part (3): Theorem 3.3(1) applied to a club of $\lambda$ consisting of singular ordinals. Part (4): Theorem 3.3(4), with $\kappa=\aleph_{j+1}$ for regular $\kappa<\aleph_{\omega_1}$ and $g_\kappa(w)=h_j(\lvert w\rvert)$ for a one-to-one map $h_j$ from $\omega$ onto $j+1$.

Dependencies

Theorem 3.3 (pp. 371--372) and Definition 3.4 (p. 372) of the same paper; the coloring follows the proof of Theorem 3.1.

Bears on

No Erdős problem in the corpus. These are negative relations at cardinals other than $\aleph_1$; Problem 474 concerns $2^{\aleph_0}\not\to[\aleph_1]^2_3$.