We report on recent progress in the ongoing study on the interaction between guessing and compactness principles.
It is well known that certain compactness principles imply the existence of diamonds. A long-standing open problem in the area asks if a weakly compact cardinal must carry a diamond sequence. We introduce a weak form of the diamond principle given in terms of function estimates on products of cardinals. We use the weaker principle to find new methods for forcing the failure of diamonds at inaccessible, Mahlo, and stationary reflecting cardinals, and show that the weaker principle must hold at a weakly compact cardinal. This is joint work with Jing Zhang.