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:
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 $S$?