Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa$, every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This is a strong form of stationary reflection.

Is it consistent that for every stationary $S \subseteq S^{\omega_3}_\omega$, there is $\alpha$ of cofinality $\omega_2$ and a club $C \subseteq \alpha$ such that $S \cap \alpha \supseteq C \cap S^{\omega_3}_\omega$?

Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa>\omega_1$, every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This is a strong form of stationary reflection.

Is it consistent that for every stationary $S \subseteq S^{\omega_3}_\omega$, there is $\alpha$ of cofinality $\omega_2$ and a club $C \subseteq \alpha$ such that $S \supseteq C \cap S^{\omega_3}_\omega$?

Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa$ and every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This is a strong form of stationary reflection.

Is it consistent that for every stationary $S \subseteq S^{\omega_3}_\omega$, there is $\alpha$ of cofinality $\omega_2$ and a club $C \subseteq \alpha$ such that $S \cap \alpha \supseteq C \cap S^{\omega_3}_\omega$?

Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa$, every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This is a strong form of stationary reflection.

Is it consistent that for every stationary $S \subseteq S^{\omega_3}_\omega$, there is $\alpha$ of cofinality $\omega_2$ and a club $C \subseteq \alpha$ such that $S \cap \alpha \supseteq C \cap S^{\omega_3}_\omega$?