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.



Cofinal rectangles in poset
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 $\omega\_1+1$.


