In Woodin's book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal" Remark 2.55 (5), it states SRP by Todorcevic (defined below) is consistent with the existence of a Suslin tree (hence it does not imply MM). Is there any reference about this? My guess is we could probably get a restricted version of MM for a Suslin tree, like the one for PFA. More precisely, $PFA(S)$ is the forcing axioms for the class of forcings that preserve $S$. Then maybe we can derive SRP from $MM(S)$.

Definition (projectively stationary sets): Let $\lambda\geq \omega_1$, then $S\subset [H(\lambda)]^\omega$ is projectively stationary if for any stationary $T\subset \omega_1$, $\{X\in S: X\cap \omega_1\in T\}$ is stationary in $[\lambda]^\omega$.

Definition (SRP): SRP asserts for any projectively stationary $S\subset [H(\lambda)]^\omega$ for regular $\lambda\geq \omega_2$, there exists a continuous increasing $\in$-chain $\langle N_\alpha: \alpha<\omega_1\rangle$ such that $N_\alpha\in S$ for all $\alpha<\omega_1$.


This appears to be proven in the following paper of Miyamoto:

Miyamoto, Tadatoshi, On iterating semiproper preorders, J. Symb. Log. 67, No. 4, 1431-1468 (2002). ZBL1050.03034.

In section 5 of the paper, he introduces a forcing axiom, SPFA(Souslin), which is the forcing axiom for semi-proper posets which preserve every Souslin tree. He shows that SPFA(Souslin) is compatible with the existence of a Souslin tree and that SPFA(Souslin) implies SRP.


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