We extend the work of Abe's 2005 paper, to show that the strong partition relation C → n+2n+1 -reg , for every club set C ⊂ Pkappalambda, is a consequence of the existence of an n-subtle cardinal. We then build on Kanamori's result that the existence of an n-subtle cardinal is equivalent to the existence of a set of ordinals containing a homogeneous subset of size n + 2 for each regressive coloring of n + 1-tuples from the set. We use this result to show that a seemingly weaker relation on the structure Pkappalambda is also equivalent. This relation is a new type of regressive partition relation, which we then almost completely characterize.;In his seminal study of the ineffability properties of cardinals, James Baumgartner discovered that the n-ineffable subsets of a cardinal kappa can be characterized as the result of "composing" the related property of n-subtlety with the classical property of P12 -indescribability. An analogous characterization was achieved by Kanamori with versions of these properties stronger than Vopenka's Principle. We generalize these theorems to show how a large class of large cardinal axioms can be composed with indescribability, and find a new instance using the Pkappa lambda versions of strongly n-ineffable and strongly n-subtle, introduced recently by Abe. In the final section, we show that each notion of n-subtlety is itself characterized as a (slightly different) kind of "composition", this time of stationary sets and what we call "pre-n-subtle" sets.
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