Abstract We establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinalμ, we have if and only if for arbitrarily largeθ<μ.

Abstract We formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle Pr1(μ+,μ+,μ+, cf (μ)) for singularμ.

Abstract We prove that if holds for a singular cardinalμ, then any collection of fewer than cf(μ) stationary subsets ofμ+must reflect simultaneously.

Abstract In this paper, we investigate the extent to which techniques used in [10], [2], and [3]—developed to prove coloring theorems at successors of singular cardinals of uncountable cofinality—can be extended to cover the countable cofinality case.

Abstract We investigate the effect of a variant of Matet forcing on ultrafilters in the ground model and give a characterization of those P–points that survive such forcing, answering a question left open by Blass [4]. We investigate the question of when this variant of Matet forcing can be used to diagonalize small filters without destroying P–points in the ground model. We also deal with the question of generic existence of stable ordered-union ultrafilters.