# 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 $S$?

## 1 Answer

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 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 $\lambda > \omega_1$.