We show that if Part(κ, λ) holds for every λ ≥ κ, then κ is strongly compact. Let κ be a regular infinite cardinal, and let λ ≥ κ be a cardinal. Pκ(λ) denotes the set of all subsets of λ of size less than κ. Part(κ, λ) means that for every F : Pκ(λ) × Pκ(λ) → 2, there is a cofinal subset A of (Pκ(λ),⊆) such that F is constant on the set {(a, b) ∈ A × A : a ⊂ b}. This definition is due to Jech [...

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: (a) If m ≥ 1 and the ultrafilter D is (im(λ),im(λ))regular then D is κ-decomposable for some κ with λ ≤ κ ≤ 2 (Theorem 4.3(a)). (b) If λ is a strong limit cardinal and D is (im(λ ),im(λ ))regular then either D is (cf λ, cf λ)-regular or there are arbitrarily large κ < λ for which D is κ-decom...

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences. One of the striking features of weak compactness in the large cardinal hierarchy is the wide variety of characterizations of the concept. These characterizations meet many disparate areas of set theory, including me...

Let H 0(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u ∈ ω∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction...

The aim of this paper is to consider questions concerning the possible maximum cardinality various separable pseudoradial (in short: SP) spaces. most intriguing question here if there in ZFC a regular (or just Hausdorff) SP space >c. While left open, we establish number non-trivial results that list below. It consistent with MA+c=ℵ2 countably tight and compact 2c. If κ measurable cardinal then ...

Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 Σ11-definable over H(κ) if there is a first order sentence φ and a parameter R ⊆ H(κ) such that functions f, g ∈ κ2 are equivalent iff for some h ∈ κ2, the structure 〈H(κ),∈, R, f, g, h〉 satisfies φ, where ∈, R, f , g, and h are interpretations of the symbols appearing in φ. All the values μ, 1 ≤ μ...

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ : For weakly compact κ there is no strongest extension of Lκω with the (κ, κ)-compactness property and the Löwenheim-Skolem theorem down to κ . With an additional set-theoretic assumption, there is no strongest extension of Lκκ with the (κ, κ)-compactness property and the Löwenhe...

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ: For weakly compact κ there is no strongest extension of Lκω with the (κ, κ)compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of Lκκ with the (κ, κ)-compactness property and the LöwenheimS...

We find many conditions equivalent to the modeltheoretical property λ κ ⇒ μ introduced in [L1]. Our conditions involve uniformity of ultrafilters, compactness properties of products of topological spaces and the existence of certain infinite matrices. See Part I [L7] or [CN, CK, KM, KV, HNV] for unexplained notation. According to [L1], if λ ≥ μ are infinite regular cardinals, and κ is a cardina...

We analyze the effect of singularizing cardinals on square properties. By work of Džamonja-Shelah and of Gitik, if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then κ,ω holds in the bigger model. We extend this to the situation where every regular cardinal in an interval [κ, ν] is singularized, for some regular cardinal ν. More precisely, we s...