On the Spectrum of Characters of Ultrafilters

Abstract

We show that the character spectrum Sp χ (λ) (for a singular cardinal λ of countable cofinality) may include any prescribed set of regular cardinals between λ and 2 λ . Nous prouvons que Sp χ (λ) (par un cardinal singulier λ avec cofinalitè nombrable) peut comporter tout l'ensemble prescrit de cardinaux reguliers entre λ et 2 λ .

Cardinal characteristics at $$\aleph_\omega$$

“…1.3 Remark that if µ is a strong limit cardinal which is our main interest, then 2 κ < µ < λ and hence every ultrafilter D over κ is generated by less than λ many sets. Observe also that actually Ch(U ) = λ in the above construction, a fact which can be proved as done in [4]. By the forcing construction of the next section it follows that one can increase 2 ℵω to some regular τ < ℵ ω 4 and obtain Sp χ (ℵ ω ) ⊇ Reg ∩ [ℵ ω+1 , τ ].…”

“…By the forcing construction of the next section it follows that one can increase 2 ℵω to some regular τ < ℵ ω 4 and obtain Sp χ (ℵ ω ) ⊇ Reg ∩ [ℵ ω+1 , τ ]. This gives a positive answer to a question from [4].…”

“…One can collapse this configuration to ℵ ω , while keeping the pcf structure. However, normal ultrafilters do not exist over the ℵ n s. Since normality plays a key-role in the above construction, it is not clear whether the inequality u λ < 2 λ is obtainable where λ = ℵ ω , and this has been asked in [4].…”

Open and solved problems concerning polarized partition relations

“…By the constructions of [22],[23], and mainly [24]. For a detailed proof we refer also to [10] and [15]. 4.2 Applying the above theorem to our problems, yields the following: ( ) The strong relation 2 µ µ → 2 µ µ 1,1 2 holds in V P .…”

Ultrafilters on singular cardinals of uncountable cofinality

“…Then there are cardinal-preserving generic extensions in which cf(κ) = λ, 2 κ is arbitrarily large and u(κ) = κ + . • (Garti, Magidor and Shelah [5,Theorem 9]) Let κ be strong, 1 let GCH hold and let µ i : i < j be an increasing sequence of measurable cardinals above κ. Let χ i : i < j be an increasing sequence of regular cardinals above κ with χ i ≤ µ i < χ i+1 .…”