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.



What are the typical forcings to shoot a club through a stationary subset of $[\lambda]^\omega$
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 $P\_{\kappa}(\lambda)$](https://open.bu.edu/bitstream/handle/2144/14099/Zaigralin_bu_0017E_10097.pdf?sequence=1) by Ivan Zaigralin. In particular look at chapter 2 of the thesis.


