# Size of stationary sets

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.

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?
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 $\lambda^{< \kappa}$)
• Also by simple-minded induction you can get the existence of stationary subsets of $P_{\omega_1}(\omega_n)$ of size $\aleph_n$ when $n\in \omega$ (of course this is trivially implied by Shelah's result). May 31, 2017 at 16:51
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.