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The following theorem is well known:

Theorem: $(\aleph_{\omega + 1}, \aleph_{\omega}) \not\twoheadrightarrow (\aleph_{n + 1}, \aleph_n)$ for every $n \geq 3$. Under CH, $(\aleph_{\omega + 1}, \aleph_{\omega}) \not\twoheadrightarrow (\aleph_{n + 1}, \aleph_n)$ for every $n > 0$.

where $(\kappa, \lambda)\twoheadrightarrow (\mu, \nu)$ stands for Chang's Conjecture between the pair of cardinals $(\kappa, \lambda)$ and the pair $(\mu, \nu)$.

Who was the first to prove this statement? Was it published somewhere?

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  • $\begingroup$ I don't know if this is the original source or not (or whether you're still interested after 5 years . . . ), but this appears as part (4) of Corollary 3.21 in this paper of Kojman, Milovich, and Spadaro: dkmj.org/academic/nt.product.pdf. $\endgroup$ Commented Oct 18, 2021 at 23:27

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Levinski+Magidor+Shelah, Chang's conjecture for $\aleph_ \omega$ -- Israel J Math 69 (1990) 161-172.

See also "Some consequences of reflection on the approachability ideal" by Matteo Vilale, Assaf Sharon, Transactions of the American Mathematical Society 362, 4201-4212, 2009. In particular fact 4.2 and remarks after it.

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    $\begingroup$ It seems like the case of CH is covered by the first paper, but the general case is a combination of the "Very weak square" of Magidor and Foreman, its connection to good points in scales (which is due to Cummings?) and Shelah's PCF argument that club many points of cofinality $\geq \aleph_4$ are good. $\endgroup$ Commented Sep 30, 2016 at 12:01

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