Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ+ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which A = ∅. The first of these cont...

We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions. For instance, we show: Theorem. Let κ be uncountable such that μω < κ for every μ < κ. If every AEC with Löwenheim-Skolem number less than κ is < κ-tame, then κ is almost strongly compact. This is done by isolating a class of AECs that exh...

Using finite support iteration of c.c.c. partial orders we provide a model of b = κ < s = κ for κ an arbitrary regular, uncountable cardinal.

Given a topological groupG (usually compact abelian), the authors study the poset D = D(G) of dense subgroups of G and its impact on the algebraic structure of G. A key tool for this is the subgroup den(G) := ∩ D. Definition. For a cardinal κ ≥ 1, a topological group is in the class Ff (κ) [F(κ); F2(κ); Fad(κ), respectively] if some family of κ-many dense subgroups of G is independent and freel...

We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “ZF + ¬ACω + Every successor cardinal is regular + E...

We prove four theorems concerning the number of normal measures a non-(κ+ 2)-strong strongly compact cardinal κ can carry.

The continuum function F on regular cardinals is known to have great freedom; if α, β are regular cardinals, then F needs only obey the following two restrictions: (1) cf(F (α)) > α, (2) α < β → F (α) ≤ F (β). However, if we wish to preserve measurable cardinals in the generic extension, new restrictions must be put on F . We say that κ is F (κ)-hypermeasurable if there is an elementary embeddi...

Using Koszmider’s strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ ≤ λ, κ = κ and 2 = κ, and η is an ordinal with κ ≤ η < κ and cf(η) = κ. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra B such that ht(B) = η + 1, wdα(B) = κ for every α < η and wdη(B) = λ (i.e. there is a local...

We study the approachability ideal I[κ+] in the context of large cardinals and properties of the regular cardinals below a singular κ. As a guiding example consider the approachability ideal I[אω+1] assuming that אω is a strong limit. In this case we obtain that club many points in אω+1 of cofinality אn for some n > 1 are approachable assuming the joint reflection of countable families of stati...

In the paper we prove that, for κ∈(0,8), n-point boundary Green’s function of exponent 8 κ−1 chordal SLEκ exists. We also that convergence is uniform over compact sets and continuous. give up-to-constant bounds function.