Suppose $\kappa$ is a regular cardinal. Does there necessarily exist a poset $\mathbb P$ that collapses $\kappa^+$ while preserving all other cardinals?



minimal collapsing without GCH
For $\kappa=\aleph\_1,$ the answer is yes, and is due to Abraham: "[On Forcing Without the Continuum Hypothesis](http://www.jstor.org/stable/2273457?seq=1#page_scan_tab_contents)". See also Todorcevic's paper "[A note on the Proper Forcing Axiom](http://www.math.toronto.edu/~stevo/Todorcevic_Note_On_PFA.pdf)", for a different proof.


For $\kappa=\aleph\_2,$ the answer is yes, and is due to Aspero: "[A forcing notion collapsing $\aleph\_3$ and preserving all other cardinals](https://www.uea.ac.uk/~bfe12ncu/collapsing-aleph3.pdf)".


For other cardinals, the answer seems to be open.


