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Suppose $(P, <)$ is a poset of cofinality $\aleph_2$ and additivity (least cardinality of an unbounded subset) $\aleph_1$. Can we conclude the existence of a cofinal subset of order-type $\omega_1 \times \omega_2$? Are there special conditions under which we get that conclusion? What about higher cofinalities and additivities? Note that the converse (if there is such a cofinal subset then cofinality and additivity are exactly those) is easy to prove.

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Let $P$ be the set of all countable subsets of $\omega_2$ ordered by set inclusion. If $2^{\aleph_0}\le\aleph_2$ then $P$ has cofinality $\aleph_2$ and additivity $\aleph_1$ and contains no chain of order type $\omega_1+1$.

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  • $\begingroup$ What about the eventual domination order on ${}^\omega \omega $? $\endgroup$ May 1, 2023 at 9:29

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