Diamonds at Large Cardinals
It is well-known that large cardinal properties can give rise to guessing principles. For example, Jensen and Kunen showed that if kappa is a subtle cardinal, then the diamond principle holds at kappa and Laver showed that if kappa is a supercompact cardinal, then a stronger guessing principle, nowadays known as the Laver diamond, holds at kappa. The following two questions are long-standing and they motivate much of the research in the area: 1) can diamond fail at a weakly compact cardinal, 2) does GCH imply that diamond holds at an inaccessible cardinal. We will introduce weakenings of the diamond principle, which follow from weak compactness. Using these principles, we demonstrate a few ways to make diamond fail at an inaccessible cardinal as well as obstacles to make diamond fail at a weakly compact cardinal. Joint work with Omer Ben-Neria.