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.

A polyadic space is a Hausdorff continuous image of some power of the onepoint compactification of a discrete space. We prove a Ramsey-like property for polyadic spaces which for Boolean spaces can be stated as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint. One corollary is that (ακ) is not a universal preimage for uniform ...

Let κ, λ be regular uncountable cardinals such that λ > κ+ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with s(κ) = λ starting from a ground model in which o(κ) = λ and prove that assuming ¬0¶, s(κ) = λ implies that o(κ) ≥ λ in the core model.

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We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ ≥ κ+ if there is an embedding j : V → M with critical point κ such that H(μ)V is included in M and j(κ) > μ. Such j is called a witnessing embedding. Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then th...

This paper develops the model theory of ordered structures that satisfy the analogue, REF (L), of the reflection principle of ZermeloFraenkel set theory. Here L is a language with a distinguished linear order <, and REF (L) consists of the universal closure of formulas of the form ∃x∀y1 < x · · · ∀y1 < x φ(y1, · · ·, yn) ↔ φ(y1, · · ·, yn), where φ(y1, · · ·, yn) is an L-formula, φ is the L-for...