TITLE: What are the Martin's Maximum consequences of Namba forcing?
QUESTION [6 upvotes]: It is known that Namba forcing is stationary-preserving and hence can be used in the setting of Martin's Maximum. Does this result in any striking consequences?

REPLY [2 votes]: Justin Moore asked a similar question a while ago, and I pointed him to my papers with Claverie and Doebler. Even though it doesn't exactly answer your question: Namba-like forcings (in the sense that they are stationary set preserving and make $\omega_2$ $\omega$-cofinal) have many applications in the presence of ${\sf MM}$. The most recent one is my proof with David Asperó that ${\sf MM}^{++}$ implies Woodin's $(*)$, see https://ivv5hpp.uni-muenster.de/u/rds/MM_implies_star.pdf .