# Two-cardinal diamond principles and saturation of the nonstationary ideal

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 $S$?

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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 $\lambda > \omega_1$.