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Matsubara–Usuba [13] introduced the notion of skinniness and its variants for subsets of P κ λ and showed that the existence of skinny stationary subsets of P κ λ is related to large cardinal properties of ideals over P κ λ and to Jensen's diamond principle on λ. In this paper, we study the existence of skinny stationary sets in more detail.

We show that if CH holds and either (i) there exists an ω 1 -Kurepa tree, or (ii) □ ( ω 2 ) holds, then there are regular T 1 Lindelof spaces X 0 and X 1 with points G δ such that the extent of X 0 × X 1 is strictly greater than 2 ω .

In this paper we investigate more characterizations and applications of δ-strongly compact cardinals. We show that, for a cardinal κ, the following are equivalent: (1) κ is δ-strongly compact, (2) For every regular λ ≥ κ there is a δ-complete uniform ultrafilter over λ, and (3) Every product space of δ-Lindelof spaces is κ-Lindelof. We also prove that in the Cohen forcing extension, the least ω 1 -strongly compact cardinal is an exact upper bound on the tightness of the products of two countably tight spaces.

We prove that for a compact Hausdorff space X, if λ c ( X ) w ( X ) for every infinite cardinal λ w ( X ) and λ c ( X ) cf ( w ( X ) ) for every infinite cardinal λ cf ( w ( X ) ) , then Tikhonov cube [ 0 , 1 ] w ( X ) is a continuous image of X, in particular the cardinality of X is just 2 w ( X ) . As an application of this result, we consider elementary submodel spaces and improve Tall's result in [17] .

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.

In this paper, we study some connections between characters of countably tight spaces of size ω 1 and inaccessible cardinals. A countable tight space is indestructible if every σ-closed forcing notion preserves countable tightness of the space. We show that, assuming the existence of an inaccessible cardinal, the following statements are consistent: (1) Every indestructibly countably tight space of size ω 1 has character ⩽ ω 1 . (2) 2 ω 1 > ω 2 and there is no countably tight space of size ω 1 and character ω 2 . For the converse, we show that, if ω 2 is not inaccessible in the constructible universe L, then there is an indestructibly countably tight space of size ω 1 and character ω 2 .

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.

We investigate the partition property of $${\mathcal{P}_{\kappa}\lambda}$$ . Main results of this paper are as follows: (1) If ? is the least cardinal greater than ? such that $${\mathcal{P}_{\kappa}\lambda}$$ carries a (? ? , 2)-distributive normal ideal without the partition property, then ? is $${\Pi^1_n}$$ -indescribable for all n < ? but not $${\Pi^2_1}$$ -indescribable. (2) If cf(?) ? ?, then every ineffable subset of $${\mathcal{P}_{\kappa}\lambda}$$ has the partition property. (3) If cf(?) ? ?, then the completely ineffable ideal over $${\mathcal{P}_{\kappa}\lambda}$$ has the partition property.

Let A be a non-empty set. A set is said to be stationary in if for every f: [A]<ω → A there exists x ϵ S such that x ≠ A and f“[x]<ω ⊆ x. In this paper we prove the following: For an uncountable cardinal λ and a stationary set S in , if there is a regular uncountable cardinal κ ≤ λ such that {x ϵ S: x ∩ κ ϵ κ} is stationary, then S can be split into κ disjoint stationary subsets.

We study various partition properties on P κ ( λ ) . Our main result asserts that if λ λ κ = λ κ , then ( p ( NSh κ , λ κ ) ) + → ( NSS κ , λ + ) 2 , where p : P κ ( λ κ ) → P κ ( λ ) is defined by p ( x ) = x ∩ λ .

Starting with a λ -supercompact cardinal κ , where λ is a regular cardinal greater than or equal to κ , we produce a model with a stationary subset S of P κ λ such that NS κ λ | S , the ideal generated by the non-stationary ideal NS κ λ over P κ λ together with P κ λ ∖ S , is λ + -saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH). We also show that in our model we can make NS κ λ | S ( κ , λ ) λ + -saturated, where S ( κ , λ ) is the set of all x ∈ P κ λ such that ot ( x ) , the order type of x , is a regular cardinal and x is stationary in sup ( x ) . Furthermore we construct a model where NS κ λ | S ( κ , λ ) is κ + -saturated but GCH fails. We show that if S ∖ S ( κ , λ ) is stationary in P κ λ , then S can be split into λ many disjoint stationary subsets.

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ⩽ ℵ 2 . We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelof if all of its subspaces of cardinality ⩽ ℵ 1 are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ⩽ ℵ 1 are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R. We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).