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

抄録

We extend the work of Abe in [1], to show that the strong partition relation C → (n+2)ⁿ⁺¹_{< -reg}, for every C ∈ WNS^*_κ,λ, 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_κλ is also equivalent. This relation is a new type of regressive partition relation, which we then attempt to characterize.