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We study the subtlety of a cardinal κ and of ℘κλ. We show that, under a certain large cardinal assumption, it is consistent that ℘κλ is subtle for some λ>κ but κ is not subtle, and the consistency of such a situation is much stronger than the existence of a subtle cardinal. We show that the subtlety of ℘κλ can be characterized by a certain partition relation on ℘κλ. We also study the property of faintness of κ, and the subtlety of ℘κλ with the strong inclusion.

For an uncountable cardinal κ, let be the assertion that every ω1-stationary preserving poset of size is semiproper. We prove that is a strong principle which implies a strong form of Chang's conjecture. We also show that implies that is presaturated.

We study combinatorial large cardinal properties on , such as ineffability, almost ineffability, subtlety, and the Shelah property. We show that, even when λ > κ, the almost ineffability of does not yield the ineffability of κ. We also show that the Shelah property and the partition property of do not yield the subtlety of κ.