In the paper "Stationary reflection and the club filter", the author Masahiro Shioya says that the club filter on $P\_{\omega\_1}(\lambda)$ cannot be $2^\lambda$-saturated for $\lambda > \omega\_1$, citing Shelah's book "Nonstructure Theory" (in preparation). I have three questions:


1) Is there a published reference for this result?


2) Does the theorem apply to $P\_{\omega\_1}(\lambda) | S$ for an arbitrary stationary set $S$?


3) Does the proof go through a two-cardinal diamond principle? I.e., did Shelah prove (in ZFC) that $\lozenge\_{\omega\_1,\lambda}$ holds for $\lambda > \omega\_1$? What about $\lozenge\_{\omega\_1,\lambda}(S)$ for arbitrary stationary $S$?


I am particularly interested in the case $\lambda = 2^{\omega} = \omega\_2$. In this case $\lozenge\_{\omega\_1,\lambda}(S)$ was proved by Donder and Matet in the paper "Two cardinal versions of diamond" for stationary sets $S$ of the form $\lbrace a \in P\_{\omega\_1}(\lambda) : \sup a \in B\rbrace$ where $B \subset \lambda$ is a stationary set consisting of points of cofinality $\omega$. Does this hold for arbitrary stationary $S$?



Two-cardinal diamond principles and saturation of the nonstationary ideal
I just came across this, but probably you resolved this question long ago. In any case, to address your questions in order as asked,


1) Shioya has an [article](http://www.ams.org/mathscinet-getitem?mr=2449470) on this. Let me know if you'd like me to ask him for a pdf.


2) I believe the consistency of local $\lambda^+$-saturation of $NS\_{\omega\_1,\lambda}$ is still open for $\lambda > \omega\_1$.


3) Yes, the proof does go through proving in ZFC $\Diamond\_{\omega\_1,\lambda}$ for all $\lambda > \omega\_1$.


