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Saharon Shelah, Jörg Brendle, Saharon Shelah
2018 Transactions of the American Mathematical Society  
For a free ultrafilter U on ω we study several cardinal characteristics which describe part of the combinatorial structure of U . We provide various consistency results; e.g. we show how to force simultaneously many characters and many π-characters. We also investigate two ideals on the Baire space ω ω naturally related to U and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter. License or copyright restrictions may apply to
more » ... ribution; see https://www.ams.org/journal-terms-of-use ULTRAFILTERS ON ω-IDEALS AND CARDINAL CHARACTERISTICS 2645 described in sections 5 and 6 are interesting, not just because they shed light on the ideal coefficients studied in section 2 and 3, but also because χ and πχ play a role in the topological investigation of βω (see [vM]). Finally, in section 7 we explore the connection between the ultrafilter characteristics and the reaping and splitting numbers r and s (see §1 for the definitions). Using iterated forcing we show (Theorem 8) that a result of Balcar and Simon ([BS], see also Proposition 7.1) which says that r is the minimum of the π-characters cannot be dualized to a corresponding statement about s. The main technical device of the proof is a careful analysis of L U -names for reals where U is a Ramsey ultrafilter. We close with a list of open problems in section 8. All sections of this work from section 2 onwards depend on section 1, but can be read independently of each other; however, §3 uses the basic definitions of §2, and sections 5 and 6 are closely intertwined. Notational remarks and some prerequisites. We refer to standard texts like [Je] or [Ku] for any undefined notion. c stands for the cardinality of the continuum. cf (κ) is the cofinality of the cardinal κ. means for all but finitely many n, and ∃ ∞ n is used for there are infinitely many n. [ω] ω ([ω] <ω , respectively) denotes the infinite (finite, resp.) subsets of ω; ω ↑ω (ω ↑<ω , resp.) stands for the strictly increasing functions from ω to ω (for the strictly increasing finite sequences of natural numbers, resp.). Identifying subsets of ω with their increasing enumerations naturally identifies [ω] ω and ω ↑ω . We reserve letters like σ, τ for elements of ω <ω and ω ↑<ω , and letters like s, t for elements of [ω] <ω . is used for concatenation of sequences (e.g., σˆ n ). Given a tree T ⊆ ω <ω , we denote by stem(T ) its stem, and by [T ] := {f ∈ ω ω ; ∀n (f n ∈ T )} the set of its branches. Given σ ∈ T , we let T σ := {τ ∈ T ; τ ⊆ σ ∨ σ ⊆ τ }, the restriction of T to σ, and succ T (σ) := {n ∈ ω; σˆ n ∈ T }. For A, B ⊆ ω, we say is called a κ-tower (or tower of height κ) iff T β ⊆ * T α for β ≥ α and T has no pseudointersection. Concerning forcing, let P be a p.o. in the ground model V . P-names are denoted by symbols likeḟ ,Ẋ, ..., and for their interpretations in the generic extension V [G], we use f =ḟ [G], X =Ẋ[G]... We often confuse Boolean-valued models V P and the corresponding forcing extensions V [G] where G is P-generic over V . P is called σ-centered iff there are P n ⊆ P with n P n = P and, for all n and F ⊆ P n finite, there is q ∈ P with q ≤ p for all p ∈ F . is used for two-step iteration (e.g. P Q ). If P α ,Q α ; α < κ (where κ is a limit ordinal) is an iterated forcing construction with limit P κ (see [B] or [Je 1] for details) and G κ is P κ -generic, we let G α = G κ ∩ P α be the restriction of the generic, and V α = V [G α ] = V Pα stands for the intermediate extension. In V α , P [α,κ) denotes the rest of the iteration. C κ (where κ is any ordinal) stands for the p.o. adding κ Cohen reals. For sections 5 and 6, we assume familiarity with Easton forcing (see [Je] or [Ku]) and the ways in which it can be factored. In particular, we use that if P is ccc and Q is ω 1 -closed (in V ), then P is still ccc in V Q and Q is ω 1 -distributive in V P . Recall that a p.o. Q is called λ-distributive iff the intersection of fewer than λ open dense subsets of Q is open dense. In section 7, we shall need basic facts about club sets in ω 1 : that the
doi:10.1090/s0002-9947-99-02257-6 fatcat:rj5ava2yljgmnmt7r7msyakdoa