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What can we say about the size of stationary subsets of $P_{\kappa}(\lambda)$ for infinite cardinals $\kappa, \lambda,$ especially when $\kappa=\aleph_1.$

Please give me some references, if there are any.

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There exists a stationary subset of $P_{\omega_1} (\omega_2)$ of size $\aleph_2$. This is a result of Baumgartner and you can find a proof for this here: Why is this set stationary?

I don't dare to answer the general case as I don't know much about it. I think it is a quite complicated issue depending on several things as cardinal arithmetic and even large cardinals.

However you can generalize the proof of Solovays Splitting theorem which says that every stationary subset of a regular cardinal $\kappa$ can be split into $\kappa$-many pairwise disjoint stationary sets, to obtain that every stationary subset of $P_{\kappa} (\lambda)$ can be split into $\kappa$ many pairwise disjoint stationary sets, which gives you a lower bound for the size of stationary sets. (An obvious upper bound is of course $\lambda^{< \kappa}$)

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Shelah proved using his pcf theory that the least cardinality of a stationary subset of $P_\kappa(\lambda)$ is equal to the least cardinality of a cofinal subset of $P_\kappa(\lambda)$. See here: M. Shioya: A proof of Shelah's strong covering theorem for $P_\kappa(\lambda)$, Asian J. Math, 12(2008), 83-98.

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