  In the context of large cardinals, the classical diamond principle
Diamond_kappa is easily strengthened in natural ways. When kappa is a
measurable cardinal, for example, one might ask that a Diamond_kappa sequence
anticipate every subset of kappa not merely on a stationary set, but on a set
of normal measure one. This is equivalent to the existence of a function
l:kappa-->V_kappa such that for any A in H(kappa+) there is an embedding
j:V-->M having critical point kappa with j(l)(kappa)=A. This and similar
principles formulated for many other large cardinal notions, including weakly
compact, indescribable, unfoldable, Ramsey, strongly unfoldable and strongly
compact cardinals, are best conceived as an expression of the Laver function
concept from supercompact cardinals for these weaker large cardinal notions.
The resulting Laver diamond principles can hold or fail in a variety of
interesting ways.
