TITLE: What are the typical forcings to shoot a club through a stationary subset of $[\lambda]^\omega$
QUESTION [9 upvotes]: Let $\lambda\geq \omega_2$ be a regular cardinal and $S\subset[\lambda]^\omega$ be a stationary set. I'm looking for a property of $S$, say "shootable", such that there exists a forcing extension preserving $\lambda, \omega_1$ as cardinals that shoots a club into $S$. I've encountered ad hoc examples, but I'd really hope if there are more explicit descriptions of a stationary set being "shootable" and given that a canonically defined reasonable forcing to shoot a club through it. I'm flexible with any cardinal arithmetic assumptions.

REPLY [7 votes]: There is a notion of fat stationary subset of $P_\kappa(\lambda)$ and a natural forcing for shooting a club through it, that you can find in the thesis Fat subsets of $P_{\kappa}(\lambda)$ by Ivan Zaigralin. In particular look at chapter 2 of the thesis.