Diamond (on the regulars) can fail at any strongly unfoldable cardinal
arXiv:math/0409304
Abstract
If kappa is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which Diamond_kappa(REG) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.
25 pages