Diamond (on the regulars) can fail at any strongly unfoldable cardinal
  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. 
Joel David Hamkins (The City University of New York) and Mirna   D\v{z}amonja (University of East Anglia)