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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 $S \supseteq C \cap S^{\omega_3}_\omega$?

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  • $\begingroup$ Mm... ${}{}{}{}$ $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 10, 2018 at 14:53
  • $\begingroup$ Käsekrainer on your mind? $\endgroup$
    – Monroe Eskew
    Commented Jul 10, 2018 at 14:55
  • $\begingroup$ Well, now it is. $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 10, 2018 at 14:56
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    $\begingroup$ Is it possible for this to fail in a model, where every stationary subset of $S^{\omega_3}_\omega$ contains a club of order-type $\omega_1$? $\endgroup$
    – Not Mike
    Commented Jul 10, 2018 at 15:10
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    $\begingroup$ I think that it can at least fail in an MM model. Force with $Add(\omega_2, \omega_3)$ over a model of MM. This forcing will preserve MM but I think that the stationary set that the forcing is adding will not contain any continuous copy of $S^{\omega_2}_{\omega}$, by density arguments. $\endgroup$
    – Yair Hayut
    Commented Jul 10, 2018 at 20:55

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