Saharon Shelah. Middle Diamond. Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah. Diamonds. Proceedings of the American Mathematical Society, vol. 138 no. 6 (2010), pp. 2151–2161. Martin Zeman. Diamond, GCH and Weak Square. Proceedings of the American Mathematical Society, vol. 138 no. 5 (2010), pp. 1853–1859. The Continuum Hypothesis (CH) implies, and is in fact equi...

Let κ be a singular cardinal of countable cofinality, 〈κn : n < ω〉 a sequence of regular cardinals which is increasing and cofinal in κ. Using a scale, we define a mapping μ from ∏ n P(κn) to P(κ+) which relates tight stationarity on κ to the usual notion of stationarity on κ+. We produce a model where all subsets of κ+ are in the range of μ for some κ a singular. Using a version of the diagona...

On the existence of large subsets of [λ] <κ which contain no unbounded non-stationary subsets * Here we deal with some problems posed by Matet. The first section deals with the existence of stationary subsets of [λ] <κ with no unbounded subsets which are not stationary, where, of course, κ is regular uncountable ≤ λ. In the second section we deal with the existence of such clubs. The proofs are...

The set of all transformation monoids on a fixed set of infinite cardinality λ, equipped with the order of inclusion, forms a complete algebraic lattice Mon(λ) with 2 compact elements. We show that this lattice is universal with respect to closed sublattices, i.e., the closed sublattices of Mon(λ) are, up to isomorphism, precisely the complete algebraic lattices with at most 2 compact elements....

We prove that the additive group (E∗, τk(E)) of an L∞-Banach space E, with the topology τk(E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E∗, τk(E)) is uniformly homeomorphic to a subset of ` κ 2 for some κ. As an immediat...

We recall that for a Urysohn space X the regular diagonal degree ∆(X) of X is defined as the minimal infinite cardinal κ such that X has a regular 2010 Mathematics Subject Classification: Primary 54A25; Secondary 54D10, 54D20

We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V |= “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + f : Ω → 2 is a function”, then there is a partial ordering P ∈ V so that for...

Let κ be an arbitrary regular infinite cardinal and let C denote the set of κ-maximal cofinitary groups. We show that if GCH holds and C is a closed set of cardinals such that 1. κ ∈ C, ∀ν ∈ C(ν ≥ κ), 2. if |C| ≥ κ then [κ, |C|] ⊆ C, 3. ∀ν ∈ C(cof(ν) ≤ κ→ ν ∈ C), then there is a generic extension in which cofinalities have not been changed and such that C = {|G| : G ∈ C}. The theorem generalize...

The problems of sphere theorems in Riemannian geometry have yielded the beautiful results and the fruitful techniques for the study of global geometry (cf. [22]). The main purpose of this paper is to study sphere theorems for CAT(1)-spaces: When are CAT(1)-spaces homeomorphic to the sphere? The notion of CAT(κ)-spaces is introduced by Gromov ([11]) based on Alexandrov’s original notion, i.e., s...

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtai...