Necessary and sufficient conditions for the existence of an n-subtle cardinal

Abstract

We extend the work of Abe in [1], to show that the strong partition relation $C \rightarrow (n+2)^{n+1}_{<-reg}$, for every $C \in \mathsf{WNS}^{*}_{\kappa,\lambda}$, is a consequence of the existence of an n-subtle cardinal. We then build on Kanamori's result in [10], 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, in the context of $P_{\kappa}\lambda$ is also equivalent. This relation is a new type of regressive partition relation, which we then attempt to characterize.