Supercompact cardinals, sets of reals, and weakly homogeneous trees.

It is shown that if there exists a supercompact cardinal then every set of reals, which is an element of L(R), is the projection of a weakly homogeneous tree. As a consequence of this theorem and recent work of Martin and Steel [Martin, D. A.

Equivalence of partition properties and determinacy.

It is shown that, within L(R), the smallest inner model of set theory containing the reals, the axiom of determinacy is equivalent to the existence of arbitrarily large cardinals below Theta with the strong partition property kappa --> (kappa)(kappa).