We obtain very strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. In particular, the simplest of our results establishes that if μ is singular and , then there is a regular cardinal θ<μ such that any fewer than cf stationary subsets of must reflect simultaneously.

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 $Pr_1(\mu^+,\mu^+,\mu^+,cf(\mu))$ for singular $\mu$.

We prove a compactness theorem for pseudopower operations of the form \}\) where \\le {{\,\mathrm{cf}\,}}\). Our main tool is a result that has Shelah’s cov versus pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set A of regular cardinals for which \\) has an inaccessible accumulation point.

Abstract.We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [1], but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.

We establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinal $\mu$, we have $\mu^+\nrightarrow[\mu^+]^2_{\mu^+}$ if and only if $\mu^+\nrightarrow[\mu^+]^2_{\theta}$ for arbitrarily large $\theta < \mu$.

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.

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.

We describe a recipe for generating normal ideals on successors of singular cardinals. We show that these ideals are related to many weakenings of □ that have appeared in the literature. Our main purpose, however, is to provide an organized list of open questions related to these ideals.

We obtain an improvement of some coloring theorems from Eisworth :1216–1243, 2010), Eisworth and Shelah :1287–1309, 2009) for the case where the singular cardinal in question has countable cofinality. As a corollary, we obtain an “idealized” version of the combinatorial principle Pr1) that maximizes the indecomposability of the associated ideal.

We prove that if ${\mu ^ + } \to \left[ {{\mu ^ + }} \right]_\mu ^2 + $ holds for a singular cardinal μ, then any collection of fewer than cf(μ) stationary subsets of μ⁺ must reflect simultaneously.

We investigate a two-player game involving pairs of filters on ω. Our results generalize a result of Shelah ([7] Chapter VI) dealing with applications of game theory in the study of ultrafilters.

We investigate pseudopowers of singular cardinals and deduce some consequences for covering numbers at singular cardinals of uncountable cofinality.