TITLE: minimal collapsing without GCH
QUESTION [11 upvotes]: Suppose $\kappa$ is a regular cardinal.  Does there necessarily exist a poset $\mathbb P$ that collapses $\kappa^+$ while preserving all other cardinals?

REPLY [12 votes]: For $\kappa=\aleph_1,$ the answer is yes, and is due to Abraham: "On Forcing Without the Continuum Hypothesis". See also Todorcevic's paper "A note on the Proper Forcing Axiom", 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".
For other cardinals, the answer seems to be open.