Subtle and Ineffable Tree Properties

Now it is obvious from the usual definitions that an inaccessible κ is subtle iff every κ-tree T satisfies STP(T ), for which we shall just write κ-STP, iff the complete binary tree 2<κ satisfies STP(2<κ), and similarly for ineffability (and one can take this as a definition if unfamiliar with the concepts). By [Mit73] one can collapse a weakly compact (a Mahlo) cardinal onto ω2 such that in the resulting universe there exists no ω2-Aronszajn tree (no special ω2-Aronszajn tree), and if there are no ω2-Aronszajn trees (no special ω2-Aronszajn trees), then (κ is weakly compact)L ((κ is Mahlo)L) holds. One can do the same for subtlety and ineffability, so that the existence of a subtle or an ineffable cardinal is also equiconsistent with the truth of certain combinatorial principles for ω2. In [MS96] it is shown that if λ is the singular limit of strongly compact cardinals, then λ+-TP holds—what about λ+-STP or λ+-ITP? Furthermore the consistency of ω+ ω -TP is proved under some large cardinal assumptions, so can we get ω+ ω -STP or ω + ω -ITP here? Baumgartner showed PFA implies ω2-TP (see [Tod84, chap. 7] or [Dev83, §5]), so we would like to know if PFA also implies ω2-STP or ω2-ITP. We can further generalize these properties to get ideals similar to the approachability ideal. For example we can consider the ideal of all subsets B of κ such that some κ-tree has an antichain which has an element of height β for every β ∈ B, so that κ-STP becomes the property this is a proper ideal. One can also reduce the requirement of having a tree with an antichain to having an antichain where the initial segments are enumerated before, so that we get an ideal containing the approachability ideal. We are then led to the question if for example on ω2 these ideals can be the nonstationary ideals on cof(ω1), cf. [Mit05].