Arch. Math. Logic DOI 10.1007/s00153-014-0412-9

Mathematical Logic

Ideals on Pκ (λ) associated with games of uncountable length Pierre Matet

Received: 16 May 2013 / Accepted: 3 November 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract We study normal ideals on Pκ (λ) that are defined in terms of games of uncountable length. Keywords

Pκ (λ) · Games · I [λ] · Scale

Mathematics Subject Classification 03E05 0 Introduction Throughout the paper κ, μ and λ will denote, respectively, a regular uncountable cardinal, a regular infinite cardinal less than κ, and a cardinal greater than or equal μ to κ. For A ⊆ Pκ (λ), G κ,λ (A) denotes the following two-person game consisting of μ moves. At step α < μ, player I selects aα ∈ Pκ (λ), and II replies by playing bα ∈ Pκ (λ). The players must follow the rule that for β < α < μ, bβ ⊆ aα ⊆ bα . II  μ wins if and only if aα ∈ A. Let N G κ,λ denote the collection of all A ⊆ Pκ (λ) α<μ

μ

such that I has a winning strategy in G κ,λ (A). μ The paper is devoted to the study of N G κ,λ . The special case κ = μ+ has been considered by Dobrinen [4], Huuskonen et al. [15], Kueker [18] and Mekler and μ μ Shelah [30]. They showed that in the definition of N G κ,λ , G κ,λ could be replaced by μ another game of the same length, and they described N G κ,λ under the assumption <μ = μ. The general case was briefly considered in [19]. The present paper that μ continues [21] which dealt with the case μ = ω, our aim being to extend the results of

P. Matet (B) Laboratoire de Mathématiques, Université de Caen-CNRS, BP 5186, 14032 Caen Cedex, France e-mail: [email protected]

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[21] to the situation when μ is uncountable. More specifically, we are going to describe μ N G κ,λ in terms of better known ideals (such as the nonstationary ideal N Sκ,λ , the ideal N μ-Sκ,λ dual to the μ-club filter, or the least [λ]<μ -normal ideal), study the effect of varying κ and λ, and investigate non-saturation properties. The part of [21] that deals with the diamond principle will be continued in a separate paper [26]. There are important differences between the case μ = ω and the case μ > ω. For one thing for μ = ω, it is not necessarily true that σ <μ < κ for every cardinal σ < κ. Another, maybe more serious, issue is that trees of uncountable height do not enjoy the same nice combinatorial properties as trees of height ω. Note however that future research might uncover some situations where the assumption μ > ω makes life actually easier. One of the many charms of the combinatorial theory of Pκ (λ) is its close connection with notions of pcf theory such as approachability, covering numbers and scales. We will be especially interested in scales with many better (or just remarkably good) points. μ Let us describe some of our results. First, concerning the description of N G κ,λ : Proposition 0.1 (Propositions 4.1, 4.6 and 4.12 and Corollary 7.9) Suppose that the GCH holds and κ is not the successor of a cardinal of cofinality less than μ. Then the following hold. μ

cf(λ) (i) If cf(λ) ≥ κ, then N G κ,λ is a restriction of N Sκ,λ . μ (ii) If cf(λ) = μ, then N G κ,λ is a restriction of Iκ,λ .

<μ

μ

[λ] (iii) If cf(λ) < μ, then N G κ,λ is a restriction of N Sκ,λ , but it is not a restriction of N Sκ,λ . μ [λ]<μ (iv) If μ < cf(λ) < κ, then N G κ,λ is not a restriction of N Sκ,λ . μ

Here is another result indicating that in case cf(λ) ∈ κ\{μ}, N G κ,λ is a complicated object : Proposition 0.2 (Corollary 7.8) Suppose that cf(λ) ∈ κ\{μ} and λ<μ ≤ λ+ . Then  + μ there is h : Pκ (λ) −→ λ+ and S ∈ N Sλ++ ∩ P E μλ such that h(N G κ,λ ) = N Sλ+ | S. Proposition 0.3 (Proposition 4.3) Suppose that cov(λ, κ, κ, μ+ ) = λ, and ρ <μ < κ μ for every cardinal ρ < κ. Then N G κ,λ = N μ-Sκ,λ . μ

In particular, if κ = μ+ and μ<μ = μ, then N G κ,λ = N μ-Sκ,λ . Proposition 0.4 (Remark after Proposition 6.3) It is consistent relative to a Mahlo cardinal that there exists a regular uncountable cardinal μ such that for any cardinal μ λ > κ = (μ )+ , N G κ ,λ = N μ -Sκ ,λ . Now for the issue of weak saturation (i.e. splitting): μ

Proposition 0.5 (i) (Proposition 9.3) N G κ,λ is nowhere weakly λ-saturated. (ii) (Proposition 9.5) Suppose that either κ is the successor of a cardinal ν and μ cf(λ) ∈ κ\{cf(ν)}, or κ is a limit cardinal and cf(λ) ∈ κ\{μ}. Then N G κ,λ is nowhere weakly λ+ -saturated.

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(iii) (Lemma 9.2 and Propositions 5.2 and 5.3 ((i) and (ii))) Suppose that λ carries a good scale of length λ+ and either κ is the successor of a cardinal ν and μ cf(λ) = cf(ν), or κ is a limit cardinal and cf(λ) = μ. Then N G κ,λ is nowhere + weakly λ -saturated. Let us now turn to non-saturation. We start with the case cf(λ) ≥ κ. Proposition 0.6 (Propositions 10.2, 10.4, 10.6 and 10.8) Suppose that one of the following is verified: (a) (b) (c) (d)

λ > κ and cf(λ) ≥ κ = ν + , where cf(ν) = μ. λ = θ + , where cf(θ ) = μ. λ = θ + , where cf(θ ) = μ, and either θ <μ = θ , or ∗θ holds. λ is weakly inaccessible and for every cardinal χ < λ, χ <μ < λ. μ

Then N G κ,λ is not λ+ -saturated. There remains the case where κ ≤ cf(λ) < λ and κ is not the successor of a cardinal μ of cofinality distinct from μ. It is already known [24] that N G κ,λ is not λ+ -saturated in this case provided that μ = ω. See Sect. 2 for the statement of Shelah’s Strong Hypothesis. Proposition 0.7 (Proposition 10.11 and Corollary 10.13) Suppose that either cf(λ) > κ and Shelah’s Strong Hypothesis holds, or cf(λ) = κ, GCH holds and κ is not the μ successor of a cardinal of cofinality distinct from μ. Then N G κ,λ is not λ+ -saturated. Now for the case cf(λ) < κ: μ

Proposition 0.8 (Proposition 10.14) If cf(λ) ∈ κ\{μ}, then N G κ,λ is not λ++ saturated. μ

If cf(λ) = μ = ω, then by a result of [23] N G κ,λ is not λ++ -saturated. Proposition 0.9 (Corollary 10.19) Suppose that cf(λ) = μ > ω and Shelah’s Strong μ Hypothesis and ADSλ both hold. Then N G κ,λ is not λ++ -saturated. The paper is organized as follows. Section 1 collects basic material concerning ideals on κ and Pκ (λ) and their properties. Section 2 collects various facts concerning μ μ N G κ,λ . The main result of this section is Proposition 2.15 that asserts that N G κ,λ is μ a projection of N G μ+ ,cov(λ,κ,κ,μ+ ) . The special case when cov(λ, κ, κ, μ+ ) = λ is investigated in Sect. 3. The results of Sect. 3 are used in Sect. 4, where we attempt μ to describe N G κ,λ under the assumption that cov(λ, κ, κ, μ+ ) = λ and ρ <μ < κ for every cardinal ρ < κ. Section 5 reviews some definitions and facts concerning some objects associated with pcf theory : scales, the ideal I [ν], pseudo-Kurepa families of various kinds, etc. Section 6 is concerned with the following question : Let ν be a regular cardinal with κ ≤ ν ≤ λ, and let S be a stationary subset of E μν . Does it then μ

+

automatically hold that {a ∈ Pκ (λ) : sup(a ∩ ν) ∈ S} ∈ (N G κ,λ ) ? In Sect. 7 it is μ shown that if cf(λ) ∈ κ\{μ} and λ<μ ≤ λ+ , then h(N G κ,λ ) = N Sλ+ | S for some h : Pκ (λ) −→ λ+ and some S ∈ N Sλ++ . Section 8 is concerned with the problem μ μ of whether N G κ,λ+ is a projection of N G κ,λ in the case where cf(λ) < κ. Finally Sects. 9 and 10 are devoted, respectively, to weak saturation and saturation properties μ of N G κ,λ .

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1 Ideals This section collects basic material concerning ideals on κ and on Pκ (λ). Let X be an infinite set. An ideal on X is a collection J of subsets of X such that (i) P(A) ⊆ J for all A ∈ J , and (ii) A ∪ B ∈ J whenever A, B ∈ J . J is proper if X ∈ / J. Let J + = P(X )\J, J ∗ = {A ⊆ X : X \A ∈ J }, and J | A = {B ⊆ X : B ∩ A ∈ J } for each A ∈ J + . For a cardinal ρ and Y ⊆ P(X ), J is Y -ρ-saturated if there is no Q ⊆ J + with |Q| = ρ such that A ∩ B ∈ Y for any two distinct members A, B of Q. J is ρ-saturated (respectively, weakly ρ-saturated) if it is J -ρ-saturated (respectively, {∅}-ρ-saturated).  P(A). For cof(J ) denotes the least cardinality of any Q ⊆ J such that J = A∈Q  a cardinal ρ, J is ρ-complete if Q ∈ J for every Q ⊆ J with |Q| < ρ. Assuming that J is κ-complete but not κ + -complete, cof(J ) denotes the least cardinality of any Q ⊆ J with the property that for any A ∈ J , there is z ⊆ Q with |z| < κ such that A ⊆ z. Given a set Y and f : X −→ Y , we let f (J ) = {A ⊆ Y : f −1 (A) ∈ J }. We say that J is nowhere such and such if for any A ∈ J + , J | A is not such and such. For a regular infinite cardinal ν, Iν denotes the noncofinal ideal on ν, and N Sν the nonstationary ideal on ν. E μκ denotes the set of all limit ordinals α < κ with cf(α) = μ. For a set A and a cardinal ρ, set Pρ (A) = {a ⊆ A : |a| < ρ}. Iκ,λ denotes the collection of all A ⊆ Pκ (λ) such that {a ∈ A : b ⊆ a} = ∅ for some b ∈ Pκ (λ). An ideal J on Pκ (λ) is fine if Iκ,λ ⊆ J . By an ideal on Pκ (λ) we will always mean a fine, proper ideal on Pκ (λ). A standard argument shows that any normal Iκ,λ - λ+ -saturated (respectively, Iκ - κ + -saturated) ideal on Pκ (λ) (respectively, on κ) is λ+ -saturated (respectively, κ + -saturated). Let δ ≤ λ, and let ν be a cardinal with 2 ≤ ν < κ. An ideal J on Pκ (λ) is [δ]<ν normal if for any A ∈ J + , and any f : A −→ Pν (δ) with the property that f (a) ⊆ a for all a ∈ A, there is B ∈ J + ∩ P(A) such that f is constant on B. It is simple to see that if δ ≥ κ, then every [δ]<ν -normal ideal on Pκ (λ) is κ-complete. Proposition 1.1 (Matet et al. [28]) There exists a [δ]<ν -normal ideal on Pκ (λ) if and only if |Pν (ρ)| < κ for every cardinal ρ < κ ∩ (δ + 1). <ν

[δ] denotes the Assuming that there exists a [δ]<ν -normal ideal on Pκ (λ), N Sκ,λ smallest such ideal. An ideal J on Pκ (λ) is δ-normal, where δ ≤ λ, if it is [δ]<2 -normal. We let <2 δ = N S [δ] . Note that J is λ-normal if and only if it is normal, so N S λ = the N Sκ,λ κ,λ κ,λ nonstationary ideal N Sκ,λ . For f : Pν (δ) −→ Pκ (λ), C κ,λ f denotes the set of all a ∈ Pκ (λ) such that a ∩ ν  = ∅ and f (e) ⊆ a for every e ∈ Pν (a ∩ δ).

Proposition 1.2 (Matet et al. [28]) Suppose that κ ≤ δ, and ρ <ν < κ for every cardinal ρ < κ. Then for any A ⊆ Pκ (λ), the following are equivalent:

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[δ] (i) A ∈ N Sκ,λ .

(ii) A ∩ C κ,λ f = ∅ for some f : Pν·3 (δ) −→ Pκ (λ).

(iii) A ∩ {a ∈ C gκ,λ : a ∩ κ ∈ κ} = ∅ for some g : Pν·3 (δ) −→ P3 (λ).  [λ]<ν   [δ]<ν  <ν ≤ λ(|δ| ) . Let us now consider cof N Sκ,λ . Thus if κ ≤ δ, then cof N Sκ,λ Proposition 1.3 (Matet et al. [27]) Assume that ν ≤ cf(λ) < κ, and ρ <ν < κ for  [λ]<ν  every cardinal ρ < κ. Then for any cardinal θ with κ ≤ θ < λ, cof N Sκ,λ ≤   [ρ]<ν   sup cof N Sκ,ρ :θ ≤ρ<λ .  [λ]<ν  It follows that if λ is a strong limit cardinal with ν ≤ cf(λ) < κ, then cof N Sκ,λ ≤  [λ]<ν  <κ λ. Let us now show that there are situations where cof N Sκ,λ ≤ λ < 2 . τ τ For a cardinal τ, ∂ κ,λ (respectively, κ,λ ) denotes the smallest cardinality of any family F of functions from τ to Pκ (λ) with the property that for any g : τ−→ Pκ (λ) (respectively, g : τ −→ λ), there is F ∈ Pκ (F) such that g(α) ⊆ f (α) f ∈F  f (α)) for every α < τ . (respectively, g(α) ∈ f ∈F

Proposition 1.4 (Matet et al. [27], Matet and Shelah [29]) Assume that δ ≥ κ, and  [δ]<ν  |δ|<ν ≤ ∂ κ,λ . ρ <ν < κ for every cardinal ρ < κ. Then cof N Sκ,λ τ

τ

Proposition 1.5 Let τ ≥ κ be a cardinal. Then ∂ κ,λ = κ,λ . τ

τ

Proof It is immediate that κ,λ ≤ ∂ κ,λ . Let us show the reverse inequality. Set X = (τ ×κ) Pκ (λ). For F ∈ Pκ (X ), define tF : τ × κ −→ Pκ (λ) by tF (γ , ξ ) =  f (γ , ξ ). Fix F ⊆ X with the property that for any g : τ × κ −→ λ, there is f ∈F

F ∈ Pκ (F) such that g(γ , ξ ) ∈ tF (γ , ξ ) for every (γ , ξ ) ∈ τ ×κ. For ( f, k) ∈ F × F, define s f k : τ −→ Pκ (λ) by s f k (γ ) = {k(γ , 1 + η) : η < sup(κ ∩ f (γ , 0))}. For a ∈ Pκ (λ), pick a bijection ja : |a| −→ a. Now let h : τ −→ Pκ (λ). Define g : τ × κ −→ λ by • g(γ , 0) = |h(γ )|. • g(γ , 1 + η) = jh(γ ) (η) if η < g(γ , 0), and g(γ , 1 + η) = 0 otherwise. Select F ∈ Pκ (F) so that g(γ , ξ ) ∈ tF (γ , ξ ) for every (γ , ξ ) ∈ τ × κ.  s f k (γ ). Claim Let γ < τ . Then h(γ ) ⊆ ( f,k)∈F ×F

Proof of the claim Let α ∈ h(γ ). There must be η < g(γ , 0) such that α = g(γ , 1 + η). Pick f and k in F so that g(γ , 0) ∈ f (γ , 0) and g(γ , 1 + η) ∈ k(γ , 1 + η). Then 

clearly, α ∈ s f k (γ ), which completes the proof of the claim. Proposition 1.6 Let σ be a regular infinite cardinal such that ν ≤ σ = 2<σ < κ, π be a cardinal greater than κ, and P be the notion of forcing that adds π Cohen subsets  λ<ν V P  λ<ν V = κ,λ . of σ . Then κ,λ

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Proof P preserves all cardinals. It is σ -closed, so for any infinite cardinal θ < σ, P  V P  V adds no new functions from θ to V . Hence, Pν (λ) = Pν (λ) , and setting  V V P  ρ = λ<ν , ρ = λ<ν . Furthermore, P satisfies the σ + -chain condition, so the following holds: (*) Suppose A, B ∈ V , and h is, in V P , a function from A to B. Then there is H : A −→ Pσ + (B) in V such that h(a) ∈ H (a) for all a ∈ A.  ρ V  ρ V P  ρ V  ρ V P ≤ ∂ κ,λ , so by Proposition 1.5 κ,λ ≤ κ,λ . Let us By (∗), κ,λ now prove the reverse inequality. Set  V • D = Pκ (λ) .  V • K = ρD .  ρ V P • τ = κ,λ . V  • Q = Pκ (τ ) .  V • R = ρλ .  + V P  + V P  V P Then by (∗), D ∈ Iκ,λ , Q ∈ Iκ,τ , and for any f ∈ ρ D , there is k ∈ K such that f (α) ⊆ k(α) for all α < ρ. Hence in V P , we may find ϕ : τ −→ K and  w : R −→ Q such that g(α) ∈ ϕ(δ))(α) whenever g ∈ R and α ∈ ρ. δ∈w(g)

By (∗), there must be in V ψ : τ −→ Pσ + (K ) and W : R −→ Pσ + (Q) such that ϕ(δ) ∈ ψ(δ) for all δ < τ , and w(a) ∈ W (g) for all g ∈ R. Define tδ : ρ −→ D for  δ ∈ τ by tδ (α) = k(α), and u : R −→ Q by u(g) = W (g). It is now k∈ψ(δ)  tδ (α) whenever g ∈ R and α ∈ ρ. 

easy to see that g(α) ∈ δ∈u(g)

Thus for example if GCH holds in V, cf(λ) < κ and κ is not the successor of a cardinal of cofinality less than cf(λ), then in the generic extension obtained by adding  [λ]
2 Basic properties of N G κ,λ μ

This section collects basic facts about N G κ,λ that will be used over and over again in the remainder of the paper (often without explicit reference).

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Recall from the introduction that A ⊆ Pκ (λ) lies in N G κ,λ if and only if I has a μ winning strategy in G κ,λ (A). μ

Proposition 2.1 N G κ,λ is the collection of all A ⊆ Pκ (λ) such that II has a winning  μ  strategy in G κ,λ Pκ (λ)\A . μ

Proof Given a winning strategy σ for I in G κ,λ (A), we define a winning strategy τ for    μ  II in G κ,λ Pκ (λ)\A by τ (a0 , . . . , aα ) = σ bβ : β ≤ α , where b0 = a0 ∪ σ (∅), and bβ = aβ for 0 < β ≤ α.  μ  Conversely, given a winning strategy θ for II in G κ,λ Pκ (λ)\A , we define a winning μ strategy χ for I in G κ,λ (A) by χ (∅) = θ (∅) and for α > 0,      di , di , . . . , di . χ (dβ : β < α) = θ ∅, i<1

A ⊆ Pκ (λ) is μ-closed if

i<2

i<α



 i<μ

ai ∈ A for every increasing sequence ai : i < μ

in (A, ⊂). C ⊆ Pκ (λ) is μ-club if C is a μ-closed cofinal subset of Pκ (λ). Let N μ-Sκ,λ be the set of all B ⊆ Pκ (λ) such that B ∩ C = ∅ for some μ-club C ⊆ Pκ (λ). Lemma 2.2 (Matet [19]) N μ-Sκ,λ is a normal ideal on Pκ (λ). μ

Proposition 2.3 N μ-Sκ,λ ⊆ N G κ,λ . μ

Proof Given C ∈ (N μ-Sκ,λ )∗ , any strategy τ for II in G κ,λ (C) with the property that 

τ (a0 , . . . , aα ) ∈ {c ∈ C : aα ⊂ c} is a winning one. μ

Next we show that N G κ,λ is a normal ideal on Pκ (λ). μ

Proposition 2.4 (i) N G κ,λ is an ideal on Pκ (λ). (ii) Suppose that there exists a [λ]<ν -normal ideal on Pκ (λ), where ν ≤ μ is an μ infinite cardinal. Then N G κ,λ is [λ]<ν -normal. μ (iii) N G κ,λ is normal. μ

μ

(i) It is obvious that P(A) ⊆ N G κ,λ for any A ∈ N G κ,λ . Now suppose that μ μ A0 , A1 ∈ N G κ,λ . To show that A0 ∪ A1 ∈ N G κ,λ , select a winning strategy τ0   μ  μ  (respectively, τ1 ) for II in G κ,λ Pκ (λ)\A0 (respectively, G κ,λ Pκ (λ)\A1 ) . We   μ define a winning strategy τ for II in G κ,λ Pκ (λ)\(A0 ∪ A1 ) by τ (a0 , . . . , aα ) = τ0 (a0 , . . . , aα ) ∪ τ1 (a0 , . . . , aα ). μ ∗ (ii) Let Ae ∈ (N G κ,λ ) for e ∈ Pν (λ). Set B = {a ∈ Pκ (λ) : ∀e ∈ Pν (a) (a ∈ μ Ae )}. For e ∈ Pν (λ), select a winning strategy τe for II in G κ,λ (Ae ). We define μ a winning strategy τ for II in G κ,λ (B) by

Proof

τ (a0 , . . . , aα ) = ν ∪



 bi ,

i≤α

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where bi =



τe (ai , ai+1 , . . . , aα ).

e∈Pν (ai )

Notice that |bi | < κ since by Proposition 1.1, |Pν (ai )| < κ. (iii) Set ν = ω and apply (ii).



Next we consider some interesting sets contained in (N μ-Sκ,λ )∗ (and hence in μ ∗ (N G κ,λ ) ). The following is immediate. Proposition 2.5 {a ∈ Pκ (λ) : |a| = |a ∩ κ| ≥ μ} is μ-club. μ

Let Yκ,λ denote the set of all nonempty a ∈ Pκ (λ) such that • For any α ∈ a, α + 1 ∈ a. • For any regular infinite cardinal ρ ∈ κ\{μ}, and any increasing sequence αξ : ξ < ρ of elements of a, sup({αξ : ξ < ρ}) ∈ a ∪ {λ}.   μ It is simple to see that if a ∈ Yκ,λ , then cf sup(a ∩ θ ) = μ for any limit ordinal θ ≤ λ with κ ≤ cf(θ ) (and hence cf(o.t.(a)) ∈ {μ, cf(λ)}). μ

Proposition 2.6 Yκ,λ is μ-club. μ

μ

Proof We define a winning strategy τ for II in G κ,λ (Yκ,λ ) by τ (a0 , . . . , aα ) = μ ∪ {δ + 1 : δ ∈ aα } ∪ dα , where dα is the set of all infinite limit ordinals η < λ such μ μ ∗ + . that cf(η) = μ and sup(aα ∩ η) = η. Thus Yκ,λ lies in (N G κ,λ ) and hence in Iκ,λ Moreover, it is clearly μ-closed. 

For α < λ, let αˆ be a fixed bijection from |α| onto α. Let Cκ,λ denote the collection of all a ∈ Pκ (λ) such that 0 ∈ a ∩ κ ∈ κ and for all α ∈ a, {|α|, α + 1} ⊆ a and a ∩ α = α“(a ˆ ∩ |α|). Note that Cκ,λ is a closed unbounded subset of Pκ (λ). Lemma 2.7 (Donder and Matet [6]) Let ν be a cardinal with κ ≤ ν ≤ λ. Suppose that a, b ∈ Cκ,λ and there is d ⊆ a ∩ b ∩ ν such that for any β ∈ (a ∪ b) ∩ ν, {ξ ∈ d : β < ξ < κ · |β|+ } = ∅. Then a ∩ ν = b ∩ ν. μ

Proposition 2.8 Suppose that μ > ω and a, b ∈ Cκ,λ ∩ Yκ,λ are such that (i) a ∩ κ = b ∩κ, and (ii) sup(a ∩χ + ) = sup(b ∩χ + ) for any cardinal χ ∈ a ∪b with κ ≤ χ < λ. Then a = b. Proof Suppose otherwise, and let α be the least element of a  b. Put η = sup(a ∩ |α|+ ). Note that η = sup(b ∩ |α|+ ) since |α| ∈ a ∪ b. Select an increasing sequence ηi : i < μ of elements of a so that α ≤ η0 and sup({ηi : i < μ}) = η. We inductively define ξγ for γ < μ as follows : • ξ0 = η0 . • ξγ = sup({ξβ : β < γ }) in case γ is an infinite limit ordinal.

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• Suppose γ is a limit ordinal. Then for any n < ω, ξγ +(2n+1) = the least δ in {ηi : i < μ} such that δ > ξγ +2n , and ξγ +(2n+2) = the least δ in b such that δ > ξγ +(2n+1) . Let d = (a ∩ α) ∪ {ξγ : γ < μ is an infinite limit ordinal}. Then clearly, d ⊆ a ∩ b ∩ |α|+ and sup(d) = η. It is simple to see that for any β ∈ (a ∪ b) ∩ |α|+ , {ξ ∈ d : β < ξ < κ · |β|+ } = ∅. Hence by Lemma 2.7 a ∩ |α|+ = b ∩ |α|+ , which yields the desired contradiction. 

Thus in case ω < μ and λ < κ +κ , we may find D ∈ (N μ-Sκ,λ )∗ with the following property : Suppose that a, b ∈ D are such that sup(a ∩ κ) = sup(b ∩ κ), and sup(a ∩ χ + ) = sup(b ∩ χ + ) for every cardinal χ with κ ≤ χ < λ. Then a = b. In particular, the following holds. Corollary 2.9 Suppose ω < μ and λ < κ +ω . Then |D| = λ for some D ∈ (N μ-Sκ,λ )∗ . This is in contrast to the situation when μ = ω. In fact it was shown in [21] that in case λ > κ, |D| ≥ λℵ0 for every D ∈ (N G ωκ,λ )∗ . μ We will now consider some other games that can be associated with N G κ,λ . Let us first recall the notion of covering number. Given four cardinals ρ1 , ρ2 , ρ3 and ρ4 such that ρ1 ≥ ρ2 ≥ ρ3 ≥ ω and ρ3 ≥ ρ4 ≥ 2, cov(ρ1 , ρ2 , ρ3 , ρ4 ) denotes the least cardinality of any X ⊆  Pρ2 (ρ1 ) with the property that for any a ∈ Pρ3 (ρ1 ), there is Q ∈ Pρ4 (X ) with a ⊆ Q. Proposition 2.10 (i) (Shelah [32], pp. 85–86) Suppose that ρ3 ≤ cf(ρ2 ) = ρ2 , +ρ ω ≤ cf(ρ4 ) = ρ4 and ρ1 < ρ2 4 , then cov(ρ1 , ρ2 , ρ3 , ρ4 ) = ρ1 .  (ii) (Matet [20])  If ρ3 = cf(ρ3 ) and  ρ4 ≥ ω, then either cf cov(ρ1 , ρ2 , ρ3 , ρ4 ) < ρ4 , or cf cov(ρ1 , ρ2 , ρ3 , ρ4 ) ≥ ρ3 . (iii) (Matet [20]) If ω ≤ ρ4 ≤ cf(ρ1 ) < ρ3 = cf(ρ3 ), then cov(ρ1 , ρ2 , ρ3 , ρ4 ) > ρ1 . Shelah’s Strong Hypothesis (SSH) asserts that given two uncountable cardinals ρ and χ with ρ ≥ χ = cf(χ ), cov(ρ, χ , χ , 2) equals ρ if cf(ρ) ≥ χ , and ρ + otherwise. Proposition 2.11 (Matet [20]) Suppose that SSH holds, and let χ ≤ κ be a regular infinite cardinal. Then cov(λ, κ, κ, χ ) equals λ+ if χ ≤ cf(λ) < κ, and λ otherwise. μ

Let Z κ,λ denote the collection of all B ⊆ Pκ (λ) such that for any a ∈ Pκ (λ), there  is z ∈ Pμ+ (B) with a ⊆ z. The following is immediate.

(ii) (iii) (iv) (v)

μ

+ (i) Iκ,λ ⊆ Z κ,λ .  μ Pκ (δ) ∈ Z κ,λ . If cf(λ) ≤ μ, then δ<λ  μ If κ = ν + , where cf(ν) ≤ μ, then Pρ (λ) ∈ Z κ,λ . ρ<κ   μ If κ = μ+ , then {α} : α < λ ∈ Z κ,λ . μ cov(λ, κ, κ, μ+ ) is the least cardinality of any member of Z κ,λ .

Lemma 2.12

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μ

For A, B ⊆ Pκ (λ), the two-person game Hκ,λ (A, B) (respectively, K κ,λ (A, B)) is defined as follows. At step α < μ, player I selects aα ∈B, and II replies by playing bα ∈ B (respectively, bα ∈ Pκ (λ)). II wins just in case (aα ∪ ba ) ∈ A. α<μ

Proposition 2.13 Let A ⊆ Pκ (λ) and B ∈ μ ∗ (N G κ,λ ) .

μ Z κ,λ .

Then the following are equivalent :

(i) A ∈ μ (ii) II has a winning strategy in Hκ,λ (A, B). μ (iii) II has a winning strategy in K κ,λ (A, B). Proof (i)↔(ii) : Select h : Pκ (λ) × μ −→ B so that d ⊆

 j<μ

h(d, j) for all

d ∈ Pκ (λ), and a one-to-one function k : μ × μ −→ μ so that k(0, 0) = 0, and k(i, j) ≥ i whenever (i, j) ∈ μ × μ. μ Given a winning strategy τ for II in G κ,λ (A), we define a winning strategy σ for II μ μ in Hκ,λ (A, B) as follows. Consider a run of the game Hκ,λ (A, B) where I’s successive moves are e0 , e1 , . . . . Define aα for α < μ by • a0 = e0 .   • aα = eα ∪ aβ if α is an infinite limit ordinal. β<α

   • aα+1 = eα ∪ h τ (a0 , . . . , aα ), j . j<μ

/ ran(k), and σ (e0 , . . . , eα ) =  Given α < μ, we let σ (e0 , . . . , eα ) = eα if α ∈ h τ (a0 , . . . , ai ), j if α = k(i, j). It is simple to see that  α<μ

aα =



   eβ ∪ σ (e0 , . . . , eγ ) .

β<μ

γ <μ μ

Conversely, given a winning strategy σ for II in Hκ,λ (A, B), we define a winning μ μ strategy τ for II in G κ,λ (A) as follows. Consider a run of the game G κ,λ (A) where / ran(k), and I successively plays a0 , a1 , . . . . Given α < μ, set eα = h(a0 , 0) if α ∈ eα = h(ai , j) if α = k(i, j). Now let τ (a0 , . . . , aα ) = aα ∪ eα ∪ σ (e0 , . . . , eα ). Then,    just as above, aα = eβ ∪ σ (e0 , . . . , eγ ) . α<μ

β<μ

For the proof of (ii)↔(iii) see [26].

γ <μ



To conclude this section, we consider a few situations when our game ideal μ can be obtained as a projection of N G κ ,λ , where κ ≤ κ and λ ≤ λ . μ

μ N G κ,λ

μ

Proposition 2.14 Let ν > λ be a cardinal. Then N G κ,λ = p(N G κ,ν ), where p : Pκ (ν) −→ Pκ (λ) is defined by p(x) = x ∩ λ. strategy Proof Fix B ⊆ Pκ (λ), and let X = p −1 (B). Given a winning  σ for II in μ μ  G κ,λ Pκ (λ)\B , we define a winning strategy τ for II in G κ,ν Pκ (ν)\X by τ (x0 , . . . , xα ) = xα ∪ σ (x0 ∩ λ, . . . , xα ∩ λ).   Conversely, given a winning strategy τfor II in G κ,ν Pκ (ν)\X , we define a winμ ning strategy σ for II in G κ,λ Pκ (λ)\B as follows. Consider a run of the game  μ  G κ,λ Pκ (λ)\B where I successively plays a0 , a1 , . . . . Define xα and tα for α < μ by

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Ideals on Pκ (λ) associated with games of uncountable length

• x 0 = a0 . ∪ tα . • xα+1 = aα+1 

 tβ if α is an infinite limit ordinal. • x α = aα ∪ β<α

• tα = τ (x0 , . . . , xα ). Now let σ (a0 , . . . , aα ) = tα ∩ λ.



Proposition 2.15 Let χ be a regular cardinal with μ < χ ≤ κ, and ν be a cardinal with cov(λ, κ, κ, χ ) ≤ ν ≤ λ<κ . Then there is a one-to-one function y : ν −→ Pκ (λ) such that • For any β ∈ λ, β ∈ y(β). μ μ • Iκ,λ ⊆ y(Iχ ,ν ) and y(N G χ ,ν ) = N G κ,λ , where y : Pχ (ν) −→ Pκ (λ) is defined  y(δ). by y(x) = δ∈x

Proof PickB ⊆ Pκ (λ) with  |B| = ν such that forany a ∈ Pκ (λ), there is z ∈ Pχ (B) with a ⊆ z. Set C = t : t ∈ Pω (B)\{∅} , and select a bijection q : ν −→ C. We define a one-to-one function h : ν −→ ν as follows. For β < λ, h(β) is defined inductively. Put h(0) = the least δ < ν such that {0} ∪ q(0) ⊆ q(δ). Suppose 0 < β < λ and h  β has been constructed. We set h(β) = the least δ < ν such that {ξ, β} ∪ q(β) ⊆ q(δ), where ξ = the least element of λ that does not belong   q h(α) . In case λ < ν, pick a bijection p : ν\λ −→ ν\ran(h  λ) to α<β

and set h = (h  λ) ∪ p. Now put y = q ◦ h. Note that ran(y) = C if λ < ν, + and q(β) ⊆ y(β) for all β < ν otherwise. It easily follows that ran(y) ∈ Iκ,λ . μ Hence by Proposition 2.13, for any A ⊆ Pκ (λ), A ∈ N G κ,λ if and only if II has a  μ  winning strategy in Hκ,λ Pκ (λ)\A, ran(y) if and only if II has a winning strategy in   μ μ Hχ ,ν Pχ (ν)\y −1 (A), Pχ (ν) if and only if y −1 (A) ∈ N G χ ,ν . Finally, let A ∈ Iκ,λ . There must be u ∈ Pχ (ν) such that {a ∈ A : y(u) ⊆ a} = ∅. Then clearly {x ∈ y −1 (A) : u ⊆ x} = ∅, and hence A ∈ y(Iχ ,ν ). 

Note that if y is as in the statement of Proposition 2.15, then clearly ran(y) ∈ (N μ-Sκ,λ )∗ . Note further that the definition of y can be simplified if we no longer insist that it be one-to-one. Simply select a bijection r : ν −→ B and let y(β) = {β} ∪ r (β) if β < ν, and y(β) = r (β) otherwise.  Corollary 2.16 Let χ and ν be as in the statement of Proposition 2.15. Then min {|C| :       μ ∗ μ ∗ C ∈ N G κ,λ } ≤ min({|D| : D ∈ N G χ ,ν } . μ

∗

μ

Proof It is simple to see that for any D ∈ (N G χ ,ν ) , y“D ∈ (N G χ ,λ )∗ , where y is as in the statement of Proposition 2.15.

 μ

In particular, cov(λ, κ, κ, μ+ ) = λ implies that min({|C| : C ∈ (N G κ,λ )∗ }) ≤ μ min({|D| : D ∈ (N G μ+ ,λ )∗ }).

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P. Matet μ

3 Ug

In the previous section we considered some interesting members of (N μ-Sκ,λ )∗ . We will now consider more, but under a cardinal arithmetic assumption that cannot hold in case μ < cf(λ) < κ.  μ g(ξ ) For g : λ −→ Pκ (λ), Ug denotes the set of all a ∈ Pκ (λ) such that a = ξ ∈e

for some e ∈ Pμ+ (a). The following is immediate. μ

Proposition 3.1 Let g : λ −→ Pκ (λ). Then Ug is μ-closed. Proposition 3.2 The following are equivalent: μ

+ for some g : λ −→ Pκ (λ). (i) Ug ∈ Iκ,λ (ii) cov(λ, κ, κ, μ+ ) = λ. μ

+ . Then clearly ran(g) ∈ Proof (i)→ (ii) : Let g : λ −→ Pκ (λ) be such that Ug ∈ Iκ,λ μ + Z κ,λ . Since |ran(g)| ≤ λ, it follows that cov(λ, κ, κ, μ ) = λ. μ

(ii)→ (i) : Suppose cov(λ, κ, κ, μ+ ) = λ. Pick g : λ −→ Pκ (λ) with ran(g) ∈ Z κ,λ .  g(γ ). Select f : Pκ (λ) −→ Pμ+ (λ) so that for any a ∈ Pκ (λ), a ⊆ γ ∈ f (a) μ  μ We define a winning strategy τ for II in G κ,λ Ug by τ (a0 , . . . , aα ) = f (aα ) ∪   μ μ ∗ + g(γ ) . Thus Ug lies in (N G κ,λ ) and hence in Iκ,λ . 

γ ∈ f (aα )

μ

Thus if cov(λ, κ, κ, μ+ ) = λ, then Ug is μ-club for some g : λ −→ Pκ (λ). Proposition 3.3 Suppose that g and h are two functions from λ to Pκ (λ) such that μ μ μ μ + . Then Ug  Uh ∈ N Sκ,λ . {Ug , Uh } ⊆ Iκ,λ Proof It suffices to show that Ug \Uh ∈ N Sκ,λ . For β ∈ λ, pick eβ ∈ Pμ+ (λ) so that  h(η). Let C be the set of all a ∈ Pκ (λ) such that eβ ∪ h(β) ⊆ a for g(β) ⊆ η∈eβ

every β ∈ a. Then clearly, C is a closed unbounded subset of Pκ (λ). Let us show that μ μ μ C ∩ Ug ⊆ Uh . Thus let a ∈ C ∩ Ug . Select t ∈ Pμ+ (a) so that a = g(β). β∈t     eβ , we get a ⊆ h(β) = h(η) ⊆ a, so Then setting v = μ

a ∈ Uh .

β∈t

β∈t

η∈eβ

η∈v



κ,λ κ,λ (respectively, E ≥μ , E μκ,λ ) denote the set of all a ∈ Pκ (λ) such that (i) Let E ≤μ   cf sup(a ∩ κ) is less than or equal to (respectively, greater  to, equal  than or equal to) μ, and (ii) for any cardinal ρ ∈ a with κ ≤ ρ < λ, cf sup(a ∩ ρ + ) is less than or equal to (respectively, greater than or equal to, equal to) μ. μ

κ,λ ∈ N Sκ,λ . Proposition 3.4 Let g : λ −→ Pκ (λ). Then Ug \E ≤μ μ

+ Proof Suppose otherwise. Then we may find A ∈ N Sκ,λ ∩ P(Ug ) and a cardinal χ ≤ λ such that (i) either χ = κ, or χ is a successor cardinal greater than κ, and  (ii) for any a ∈ A, sup(a ∩ χ ) ∈ / a and cf sup(a ∩ χ ) > μ. Given a ∈ A, pick

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Ideals on Pκ (λ) associated with games of uncountable length

   da ∈ Pμ+ (a) and ea ⊆ a ∩ χ so that a = g(α), o.t.(ea ) = cf sup(a ∩ χ ) α∈da and sup(ea ) = sup(a ∩ χ ). Then we may find αa ∈ da such that |ea ∩ g(αa )| = |ea |. + There must be B ∈ N Sκ,λ ∩ P(A) and β such that αa = β for all a ∈ B. Then   sup(a ∩ χ ) = sup g(β) for all a ∈ B, which yields the desired contradiction. 

+

Proposition 3.5 Suppose λ < κ +(μ ) . Then there is g : λ −→ Pκ (λ) such that  μ κ,λ ∗ Ug ∈ N Sκ,λ | E ≤μ . Proof Set λ = κ +θ . Select a bijection j : Pω (λ) −→ λ. Recall from Sect. 2 that for each β < λ, βˆ denotes a fixed bijection from |β| onto β. Define k : Pω (λ) −→ Pκ (λ) as follows. If e = {α}, where α < κ, put k(e) = α. If e = {α0 , . . . , αn }, where αn ◦ . . . ◦ α1 )(ξ ) : 1 ≤ n < ω, α0 < κ, and αi < |αi+1 | for every i < n, set k(e) = {( ξ < α0 }. Otherwise let k(e) = ∅. Now put g = k ◦ j −1 . Let D be the set of all ∗ . a ∈ Cκ,λ such that {κ +β : β < θ } ⊆ a = { j (e) : e ∈ Pω (a)}. Note that D ∈ N Sκ,λ μ

κ,λ κ,λ Let us prove that D ∩ E ≤μ ⊆ Ug . Thus fix a ∈ D ∩ E ≤μ . Pick z 0 ⊆ a ∩ κ so that o.t.(z 0 ) = cf(a ∩ κ) and sup(z 0 ) = a ∩ κ. Furthermore for each β < θ , select  ∩ κ +β+1 ) and z β+1 ⊆ {δ ∈ a : κ +β ≤ δ < κ +β+1 } so that o.t.(z β+1 ) = cf sup(a 

 z β+1 and v = { j (e) : e ∈ sup(z β+1 ) = sup(a ∩ κ +(β+1) ). Now set t = z 0 ∪ β<θ   μ k(e) = g(γ ). Hence a ∈ Ug . Pω (t)}. It is easily verified that a = e∈Pω (t) γ ∈v 

These results will be used in the next section. μ

<μ

[λ] 4 The relation between N G κ,λ and N Sκ,λ

Assuming that cov(λ, κ, κ, μ+ ) = λ and there exists a [λ]<μ -normal ideal on Pκ (λ), μ [λ]<μ we now investigate the relation between N G κ,λ and well-known ideals such as N Sκ,λ or N Sκ,λ . Throughout the section it is assumed that there exists a [λ]<μ -normal ideal on Pκ (λ), i.e. that ρ <μ < κ for every cardinal ρ with μ ≤ ρ < κ. Let us observe that by a result of [28], if μ is the successor of a singular cardinal θ , [λ]<μ [λ]<θ and cov(λ, θ, θ, σ ) = λ for some infinite cardinal σ < θ , then N Sκ,λ = N Sκ,λ . <μ

μ

[λ] Proposition 4.1 Suppose cov(λ, κ, κ, μ+ ) = λ. Then N G κ,λ = N Sκ,λ some g : λ −→ Pκ (λ).

μ

| Ug for

Proof By Proposition 2.15, there is a one-to-one function y : λ −→ Pκ (λ) such that • β ∈ y(β) for all β ∈ λ. μ μ • N G κ,λ = y(N G μ+ ,λ ), where y : Pμ+ (λ) −→ Pκ (λ) is defined by y(x) =  y(δ). δ∈x

μ

<μ

μ

μ

[λ] | U y ⊆ N G κ,λ . Then clearly, ran(y) = U y . Hence by Proposition 2.4 (ii), N Sκ,λ  [λ]<μ + μ with A ⊆ U y . We need to show To prove the reverse inclusion, fix A ∈ N Sκ,λ μ ∗ that A ∩ B = ∅ for every B ∈ (N G κ,λ ) . Fix such a B. By Proposition 2.13, II has a

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P. Matet

 μ  winning strategy τ in Hκ,λ B, Pκ (λ) . Select a bijection k : μ × λ −→ λ, and let D

  y(δ) ⊆ a = k“(μ × a). Note that be the set of all a ∈ Pκ (λ) such that μ ∪ δ∈μ  ∗ . For a ∈ A ∩ D, pick e ∈ P + (a) so that μ ⊆ e and a = D ∈ N Sκ,λ y(ξ ), a a μ ξ ∈ea

and a : μ −→   ea . Now let T be the set of all a ∈ A ∩ D such that  a bijection τ y(a (0) , . . . , y(a (α) \a = ∅ for some α < μ. <μ

[λ] Claim T ∈ N Sκ,λ .

 [λ]<μ + Proof of the claim Suppose otherwise. Then we may find β ∈ μ and T ∈ N Sκ,λ      ∩ P(T ) such that τ y a (0) , . . . , y a (β) \a = ∅ for every a ∈ T . There must be  [λ]<μ +   d ∈ Pμ (λ) and T ∈ N Sκ,λ ∩ P(T ) such that {k α, a (α) : α ≤ β} = d for every a ∈ T . Define t :β+1 −→ λ by  t (α)  = the unique ζ ∈ λ such that k(α, ζ ) ∈ d. Pick b ∈ T so that τ y t (0) , . . . , y t (β) ⊆ b. Clearly, t  (β + 1) = b  (β + 1). This contradiction completes the proof of the claim. By the claim (A ∩ D)\T = ∅. Now for any a ∈ (A ∩ D)\T , a=

         y a (α) ∪ τ y a (0) , . . . , y a (α) α<μ

α<μ

and therefore a ∈ B.



μ

Proposition 4.1 can be used to clarify the relation between N G κ,λ and N μ-Sκ,λ . <μ

[λ] Lemma 4.2 (Dobrinen [4]) N Sκ,λ

⊆ N μ-Sκ,λ . μ

Proposition 4.3 Suppose cov(λ, κ, κ, μ+ ) = λ. Then N G κ,λ = N μ-Sκ,λ . μ

<μ

μ

[λ] | Ug . Proof By Proposition 4.1, there is g : λ −→ Pκ (λ) such that N G κ,λ = N Sκ,λ μ ∗ Then Ug ∈ (N μ-Sκ,λ ) by Proposition 3.1. Hence by Proposition 2.3 and Lemma 4.2, μ

<μ

[λ] N μ − Sκ,λ ⊆ N G κ,λ = N Sκ,λ

| Ugμ ⊆ N μ − Sκ,λ . 

Let us next turn to the special case κ = μ+ . Note that in this case cov(λ, κ, κ, μ+ ) = λ trivially holds. <μ

μ

[λ] . Proposition 4.4 Suppose κ = μ+ . Then N G κ,λ = N Sκ,λ μ

<μ

μ

[λ] | Ug . Proof By Proposition 4.1, there is g : λ −→ Pκ (λ) such that N G κ,λ = N Sκ,λ μ Let h : λ −→ Pκ (λ) be defined by h(α) = {α}. Then clearly Uh = Pκ (λ). It μ ∗ , since by Proposition 3.3 U μ  U μ ∈ N S . Hence follows that Ug ∈ N Sκ,λ κ,λ g h <μ

[λ] N Sκ,λ

μ

<μ

[λ] | Ug = N Sκ,λ .



μ

See [30] for another characterization of N G κ,λ in the case where κ = μ+ .

123

Ideals on Pκ (λ) associated with games of uncountable length μ

Assuming cov(λ, κ, κ, μ+ ) = λ, Proposition 4.1 shows that N G κ,λ is a restriction of the least [λ]<μ -normal ideal on Pκ (λ). In the remainder of the section we μ use additional assumptions to obtain N G κ,λ as a restriction of an ideal smaller than <μ

[λ] N Sκ,λ .

<μ

[λ] Lemma 4.5 (Matet et al. [28]) Suppose λ<μ = λ. Then N Sκ,λ some A.

= N Sκ,λ | A for μ

Proposition 4.6 Suppose that cov(λ, κ, κ, μ+ ) = λ = λ<μ . Then N G κ,λ = N Sκ,λ | B for some B. 

Proof By Proposition 4.1 and Lemma 4.5. μ

In particular, if λ < κ +ω , then N G κ,λ = N Sκ,λ | B for some B. Let us next see how this can be sharpened and generalized. Lemma 4.7 Suppose that λ < κ +σ , where σ is a regular infinite cardinal less than [λ]<μ [λ]<σ κ,λ μ. Then N Sκ,λ = N Sκ,λ | E ≥μ . <σ

<μ

[λ] [λ] κ,λ | E ≥μ ⊆ N Sκ,λ . To prove the reverse incluProof It is simple to see that N Sκ,λ  [λ]<σ + κ,λ ∩ P(E ≥μ ), and let ea ∈ Pμ (a) for a ∈ A. For each singular sion, fix A ∈ N Sκ,λ cardinal ν with κ < ν ≤ λ, pick an increasing sequence πiν : i < cf(ν) of successor cardinals such that κ ≤ π0ν and sup({πiν : i < cf(ν)}) = ν. Select two bijections    n (λ ∪ {λ}) −→ λ and g : λ × Pμ (α) −→ λ. Now let B be the f : n<ω α<κ set of all a ∈ A ∩ Cκ,λ such that

• • • • •

μ ⊆ a. For any cardinal ρ with κ ≤ ρ < λ, ρ ∈ a. For each singular cardinal ν with κ < ν ≤ λ, {πiν : i < cf(ν)} ⊆ a. For any n ∈ ω and any t : n −→ a ∪ {λ},  f (t) ∈ a. g(β, d) ∈ a whenever β ∈ a and d ∈ Pμ (α). α∈a∩κ

 n Note that B ∈ For a ∈ B, we will define a tree Ta ⊆ (a ∪ n<ω {λ}), and a labeling function ϕa : Ta −→ Pμ (a) with the property that for any t ∈ Ta \{∅}, ϕa (t) ⊆ t ((dom(t)) − 1). Thus let a ∈ B. For t ∈ Ta , the set {v ∈ Ta : t ⊂ v and dom(v) = 1 + dom(t)} of immediate successors of t in Ta will be denoted by sucTa (t). Put ϕa (∅) = ∅, sucTa (∅) = {t0 }, where t0 = {(0, λ)}, and ϕa (t0 ) = ea . Suppose t ∈ Ta \{∅} and ϕa (t) have been constructed. Put dom(t) = m, t (m − 1) = α and ϕa (t) = d. 

[λ]<σ + N Sκ,λ .

Case 1 α < κ. Let sucTa (t) = ∅. Case 2 α = κ or α is a successor cardinal greater than or equal to κ. Setting β = the least γ ∈ a such that d ⊆ γ , put w = t ∪ {(m, β)}, and let sucTa (t) = {w} and ϕa (w) = d.

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P. Matet

Case 3 κ ≤ |α| < α. α −1 (d). Setting v = t ∪ {(m, |α|)}, let sucTa (t) = {v} and ϕa (v) = Case 4 α is a singular cardinal greater than κ. For i < cf(α), set u i = t ∪ {(m, πiα )}. Now put sucTa (t) = {u i : i < cf(α)}, and ϕa (u i ) = d ∩ πiα for every i < cf(α). Note that Ta has no infinite branches since each t in Ta \{∅} is a decreasing function. Put [Ta ] = {t ∈ Ta : sucTa (t) = ∅}. Clearly, |sucTa (t)| < σ for every t ∈ Ta , so each level of Ta has size less than σ , and therefore |[Ta ]| < σ . For t ∈ [Ta ], set Mt = {m ∈ dom(t) : m ≥ 1 and κ ≤ |t (m − 1)| < t (m − 1)}. The following is obvious. Claim 1 Let t ∈ [Ta ] be such that Mt = ∅. Then dom(t) = 2, [Ta ] = {t} and ϕa (t) = ea . Given t ∈ [Ta ] with Mt = ∅, let m t :  ≤ jt  be the increasing enumeration of    t (ζ ) : ζ ∈ ϕ (t) . Mt . For  ≤ jt , set αt = t (m t − 1). Put xt = α 0t ◦ . . . ◦ α a jt  Claim 2 Suppose Mt = ∅ for every t ∈ [Ta ]. Then ea = xt . t∈[Ta ]

Proof of Claim 2 Given t ∈ [Ta ], it is readily checked that • For m ≤ m t0 , ϕa (t  m) ⊆ ea .

−1

−1  −1 t t   . . . α 0t • For 0 <  ≤ jt and m t−1 < m ≤ m t , ϕa (t  m) ⊆ α α −1 −2  (ea ) . . . .

 −1  t −1 α t t −1 (ea ) . . .  .  • For m tjt < m ≤ dom(t), ϕa (t  m) ⊆ α . . . α 0 jt jt −1

Hence, xt ⊆ea . That ea ⊆ xt can be easily derived from the following observation. Supt∈[Ta ] pose t ∈ Ta \[Ta ] and ξ ∈ ϕa (t). Then • If |sucTa (t)| = 1, then ξ ∈ ϕa (v) for some v ∈ sucTa (t). α (ζ ) : ζ ∈ • If sucTa (t) = {w}, then  either ξ ∈ ϕa (w), or κ ≤ |α| < α and ξ ∈ { ϕa (w)}, where α = t (dom(t) − 1 . This completes the proof of Claim 2.  [λ]<σ + We may find H ∈ N Sκ,λ ∩ P(B) and r ∈ Pσ (λ) such that {g( f (t), ϕa (t)) : t ∈ [Ta ]} = r for every a ∈ H . Now let a, b be two elements of H . Then clearly [Ta ] = [Tb ] (and consequently Ta = Tb ), and ϕa  [Ta ] = ϕb  [Tb ]. Hence by Claims 

1 and 2, ea = eb . Proposition 4.8 Suppose that λ < κ +σ , where σ is a regular infinite cardinal less μ [λ]<σ than μ. Then N G κ,λ = N Sκ,λ | E μκ,λ . Proof Since cov(λ, κ, κ, μ+ ) = λ by Proposiion 2.10 (i), there is by Proposition 4.1 μ μ [λ]<μ | Ug . Now by Propositions 3.3 and g : λ −→ Pκ (λ) such that N G κ,λ = N Sκ,λ

123

Ideals on Pκ (λ) associated with games of uncountable length ∗

μ

<σ

μ

[λ] κ,λ κ,λ 3.5, Ug ∈ (N Sκ,λ | E ≤μ ) . Moreover, Ug \E ≤μ ∈ N Sκ,λ . It follows that N Sκ,λ

μ Ug

= μ

[λ]<σ N Sκ,λ

|

κ,λ E ≤μ .

<σ

[λ] N G κ,λ = N Sκ,λ

|

Hence by Lemma 4.7,

 [λ]<σ  κ,λ [λ]<σ κ,λ  κ,λ  = N Sκ,λ = N Sκ,λ | Ugμ ∩ E ≥μ | E ≤μ ∩ E ≥μ | E μκ,λ . 

Corollary 4.9 Suppose μ > ω and λ <

κ +ω .

Then

μ N G κ,λ

= N Sκ,λ |

E μκ,λ .

Let us remark that for μ = ω, we have the following better result. μ

Proposition 4.10 (Matet [21]) Suppose that μ = ω and λ < κ +ω1 . Then N G κ,λ = N Sκ,λ | E μκ,λ . μ

Let us finally consider situations when N G κ,λ = J | A, where J ⊂ N Sκ,λ . Lemma 4.11 (Matet et al. [27], Matet and Shelah [29]) Suppose that μ ≤ cf(λ) < λ,  [ρ]<μ  and there is a cardinal θ < λ such that cof N Sκ,ρ ≤ λ for every cardinal ρ with cf(λ) [λ]<μ = N Sκ,λ | D for some D. θ ∪ κ ≤ ρ < λ. Then N Sκ,λ Proposition 4.12 (i) Suppose that cf(λ) = μ, cov(λ, κ, κ, μ+ ) = λ, and there  [ρ]<μ  is a cardinal θ < λ such that cof N Sκ,ρ ≤ λ for every cardinal ρ with μ θ ∪ κ ≤ ρ < λ. Then N G κ,λ = Iκ,λ | A for some A. (ii) Suppose that κ ≤ cf(λ) < λ = cov(λ, κ, κ, μ+ ), and there is a cardinal θ < λ  [ρ]<μ  < λ for every cardinal ρ with θ ∪ κ ≤ ρ < λ. Then such that cof N Sκ,ρ μ cf(λ) N G κ,λ = N Sκ,λ | A for some A. Proof By Proposition 4.1 and Lemma 4.11.



5 pcf theory and related objects We will now review some material that will be used in the next sections. We start with the notion of scale. Let A be an infinite set of regular infinite cardinals such that |A| < min(A), and I be a proper ideal on A. For f, g ∈ A On, let f < I g if {a ∈ A : f (a) < g(a)} ∈ I ∗ , ≤ g(a)} ∈ I ∗ . and f ≤ I g if {a ∈ A : f (a)    A h(a). We let A = h, where h(a) = a for all For h ∈ On, let h = a∈A a ∈ A.  Let h : A −→ On\{0}, and let π be a regular cardinal. We say that tcf( h/I ) = π if there  exists an < I -increasing sequence  f ξ : ξ < π  in h such that for any g ∈ h, there is ξ < π with g ≤ I f ξ . Such a sequence is said to be a scale of length π for sup(A). Proposition 5.1 (Shelah (see e.g. [1], Theorems 2.23 and 2.26)) Suppose that λ is singular. Then there is a set A of regular  infinite cardinals such that o.t.(A) = cf(λ), sup(A) = λ, |A| < min(A) and tcf( A/I ) = λ+ , where I is the noncofinal ideal on A.

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We omit the definitions of ppχ (λ) and pp(λ) which can be found in [32], p. 41. Let us now turn to pseudo-Kurepa families. Given an infinite cardinal τ ≤ κ, and a cardinal π > λ, a (τ, λ, π )-sequence is a sequence y = yα : α < π  of elements of Pτ (λ) with the property that yα = {α} for every α < λ. An Aκ,λ (τ, π )-sequence is a (τ, λ, π )-sequence y = yα : α < π  with the property that |{α < π : yα ⊆ a}| < κ for every a ∈ Pκ (λ). Aκ,λ (τ, π ) asserts the existence of an Aκ,λ (τ, π )-sequence. Proposition 5.2 (Matet [22]) Let ρ be the largest limit cardinal less than or equal to κ. Assume that cf(λ) < κ and one of the following conditions is satisfied : (a) ρ = κ. (b) cf(λ) ∈ ρ\{cf(ρ)}.   (c) cf(λ) = cf(ρ) < ρ and min pp(ρ), ρ +3 < κ.   +3  (d) cf(λ) ≥ ρ and min 2cf(λ) , cf(λ) < κ.  + +  Then Aκ,λ cf(λ) , λ holds. A Bκ,λ (τ, π )-sequence is a (τ, λ, π )-sequence y = yα : α < π  with the property yα . It is simple that for each nonempty e in Pκ + (π ), there is a < κ-to-one g ∈ α∈e to see that any Bκ,λ (τ, π )-sequence is an Aκ,λ (τ, π )-sequence. Bκ,λ (τ, π ) asserts the existence of a Bκ,λ (τ, π )-sequence. I (ν + , π )Let ν < κ be an infinite cardinal, and I be a proper ideal on ν. A Dκ,λ sequence is a (ν + , λ, π )-sequence y = yα : α < π  with the following property : there is f α : ν −→ λ for λ ≤ α < π such that (a) for any nonempty e ∈ Pκ + (π \λ), there  is g : e −→  I such that f α (i) = f β (i) whenever α < β are in e and i is in ν\ g(α) ∪ g(β) , and (b) there is a bijection ψ : ν × λ −→ λ such that for any   I (ν + , π )-sequence α ∈ π \λ, yα = {ψ i, f α (i) : i < ν}. It is easy to see that any Dκ,λ + is a Bκ,λ (ν , π )-sequence. ADSλ asserts the existence of a sequence z β : β < λ+  such that (i) for any + β < λ+ , z β is an order-type cf(λ), cofinal subset of λ, and (ii) for  any δ < λ , there is g : δ −→ λ with the property that (z β \g(β) ∩ (z γ \g(γ ) = ∅ whenever β < γ < δ. It is simple to see that if cf(λ) < κ and ADSλ holds, then there exists a  +  I Dκ,λ cf(λ) , λ+ -sequence, where I = Icf(λ) . Scales with nice properties can be used to show that these principles hold. Let us remark that Proposition 5.1 is a crucial ingredient in the proof of Proposition 5.2.   λi /I = π , where Suppose that tcf i<ν

• ν is a cardinal with cf(λ) ≤ ν < κ. • λi : i < ν is a one-to-one sequence of regular cardinals less than λ such that ν < λ0 and sup({λi : i < ν}) = λ. • I is a proper ideal on ν such that for any cardinal χ < λ, {i ∈ ν : λi ≤ χ } ∈ I.   Let f =  f α : α < π  be an increasing, cofinal sequence in λi , < I . i<ν An infinite limit ordinal δ < π is a good point for f if we may find a cofinal subset X of δ, and Z ξ ∈ I for ξ ∈ X such that f β (i) < f ξ (i) whenever β < ξ are in X and i ∈ ν\(Z β ∪ Z ξ ). δ is a better (respectively, remarkably good) point for f if we may

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find a closed unbounded subset X of δ, and Z ξ ∈ I for ξ ∈ X such that f β (i) < f ξ (i) whenever β < ξ are in X and i is in ν\Z ξ (respectively, ν\(Z β ∪ Z ξ )). δ is a very good point for f if there is a closed unbounded subset X of δ, and Z ∈ I such that f β (i) < f ξ (i) whenever β < ξ and i ∈ ν\Z . Note that very good ⇒ better ⇒ remarkably good ⇒ good. An easy argument shows that if δ is a good point for f, then cf(δ) < λ. It is shown in [2] that if I is cf(δ)-complete, then δ is a better point for f. The scale f is good (respectively, remarkably good, better, very good), if there is a closed unbounded subset C of π such that every δ ∈ C with ν < cf(δ) < λ is a good (respectively, remarkably good, better, very good) point for f. Proposition 5.3 Let C be a closed unbounded subset of π . Then the following hold. (i) (Matet [22]) Suppose that κ is a successor cardinal, and κ ≤ (ρ · ν)+3 , where ρ is the largest limit cardinal less than κ. Suppose further that there is j ∈ {1, 2, 3} π  such that (ρ · ν)+ j ≤ κ, and every δ ∈ C ∩ E (ρ·ν) + j is a good point for f . Then + Aκ,λ (ν , π ) holds. (ii) (Matet [22]) Suppose that κ is a limit cardinal, and for any cardinal τ with ν ≤ τ < κ, and any δ ∈ C ∩ E τπ+ , δ is a good point for f. Then there is X ⊆ Pν + (λ) with |X | = π such that |X ∩ P(b)| ≤ |b| for any b ∈ Pκ (λ). (iii) (Matet [25]) Suppose that every δ in C ∩ E κπ is a remarkably good point for f. Then Bκ,λ (ν + , π ) holds. (iv) (Matet [25]) Let σ ≤ ν be an infinite cardinal. Suppose that I is σ -complete, and for any regular cardinal θ with σ < θ ≤ κ and any δ ∈ C ∩ E θπ , δ is a I (ν + , π ) holds. remarkably good point for f. Then Dκ,λ (v) (Matet [25]) Let σ ≤ ν be an infinite cardinal. Suppose that I is σ -complete, and for any regular cardinal θ with σ < θ < π and any δ ∈ C ∩ E θπ , δ is a remarkably good point for f. Then for any ξ < π , there is g : C ∩ ξ −→ I with the  property that  f β (i) < f γ (i) whenever β < γ are in C ∩ ξ and i ∈ ν\ g(β) ∪ g(γ ) . Let us also mention the following (for the definition of pp(κ,μ+ ) (λ), see [32], pp. 39 and 41). Proposition 5.4 (Shelah [32], Claim 1.5A and Remark 1.5B (4) p. 51) Suppose that cf(λ) < λ and π < pp(κ,μ+ ) (λ). Then for some infinite cardinal ν < κ, and some I (ν + , π )-sequence. μ+ -complete proper ideal I on ν, there exists a Dκ,λ Let us recall the definition of the weak square principle. Given an infinite cardinal ρ, ∗ρ asserts the existence of Cα for any infinite limit ordinal α < ρ + such that (i) (ii) (iii) (iv) (v)

1 ≤ |Cα | ≤ ρ. Each C ∈ Cα is a closed unbounded subset of α. If cf(α) < ρ, then o.t. (C) < ρ for every C ∈ Cα . If C ∈ Cα and β is a limit point of C, then C ∩ β ∈ Cβ . Cα contains a set of order-type cf(α).

That (v) can be added to the usual conditions (i)–(iv) has been proved by Foreman and Magidor [9]. It is simple to see that ∗ρ follows if ρ <ρ = ρ.

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Proposition 5.5 (Cummings et al. [2]) Suppose that λ is singular and ∗λ holds. Then ADSλ holds. The next notion we are going to review is that of approachability. I [κ] denotes the collection of all S ⊆ κ for which we may find a closed unbounded subset C of κ, and z β ⊆ κ for β < κ such that for each α ∈ C ∩ S, (a) α is a limit ordinal with cf(α) < α, and (b) there is an unbounded subset d of α of order-type cf(α) with the property that {d ∩ δ : δ < α} ⊆ {z β : β < α}. The following is readily checked. Proposition 5.6 Let S ⊆ κ be such that {ρ ∈ S : ρ is a regular cardinal} ∈ N Sκ . Then the following are equivalent: (i) S ∈ I [κ]. (ii) There is a sequence dα : α ∈ S such that (a) for each infinite limit ordinal α in S, dα is a cofinal subset of α of order-type cf(α), and (b) for any T ∈ N Sκ+ ∩P(S) and any δ < κ, there is W ⊆ T such that W ∈ N Sκ+ and |{dα ∩δ : α ∈ W }| = 1. Assertion (ii) in Proposition 5.6 is due to Donder and Levinski [5] who observed that it is equivalent to the apparently weaker assertion obtained by replacing “W ∈ N Sκ+ ” by “W ∈ Iκ+ ” and “|{dα ∩ δ : α ∈ W }| = 1” by “|{dα ∩ δ : α ∈ W }| < κ”. Proposition 5.7 (i) (Shelah (see e.g. Theorem 3.18 in [7])) If μ+ < κ, then S ∈ I [κ] for some S ∈ N Sκ+ ∩ P(E μκ ). (ii) (Shelah (see e.g. Lemma 3.40 in [7])) Suppose that χ <μ < κ for every infinite cardinal χ < κ. Then E μκ ∈ I [κ]. +

(iii) (Shelah (see e.g. Corollary 4.6 in [7])) E μκ ∈ I [κ + ]. (iv) ([7], Proposition 4.11) Let ρ be an infinite cardinal such that ∗ρ holds. Then ρ + ∈ I [ρ + ].

6 sup(a ∩ ν) Let ν be a regular cardinal with κ ≤ ν ≤ λ, and let g : Pκ (λ) −→ ν be defined by μ g(a) = sup(a ∩ ν). The question we address in this section is whether g(N G κ,λ ) ⊆ ν N Sν | E μ . Proposition 6.1 Let n ∈ ω, and let ν0 < ν1 < · · · < νn be regular cardinals   such that κ ≤ ν0 and νn ≤ λ. For each i ≤ n, let Si ∈ N Sν+i ∩ I [νi ] ∩ P E μνi . Then μ + {a ∈ Pκ (λ) : ∀i ≤ n (sup(a ∩ νi ) ∈ Si )} ∈ (N G κ,λ ) . Proof We deal with the case n > 0 and leave the (simpler) proof of the result in case n = 0 to the reader. For i ≤ n, select Ci , z βi : β < νi  and dαi : α ∈ Ci ∩ Si  so that • Ci is a closed unbounded subset of νi . • For any α ∈ Ci ∩ Si , dαi is an unbounded subset of α of order-type μ with the property that {dαi ∩ δ : δ < α} ⊆ {z βi : β < α}. For i ≤ n and α ∈ Ci ∩ Si , let ξθi,α : θ < μ be the increasing enumeration of dαi .

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 μ ∗ μ  Fix A ∈ (N G κ,λ ) . Select a winning strategy τ for II in Hκ,λ A, Pκ (λ) , and a   i, f (i)  f (Ci ∩ Si ), let ek(i,θ) = ξθ bijection k : (n + 1) × μ −→ μ. Given f ∈ i≤n

f

f

f

for i ≤ n and θ < μ, and bζ = τ (e0 , . . . , eζ ) for ζ < μ. Given j ≤ n, α ∈ C j ∩ S j   j,α j,α and g ∈ (Ci ∩ Si ), we define f g ∈ (Ci ∩ Si ) by f g (i) = g(i) i∈(n+1)\{ j} j,α

i≤n

if i = j, and f g (i) = α otherwise.  j Claim Let j ≤ n and g ∈ (Ci ∩ Si ). Then there is Dg ∈ N Sν∗j such that i∈(n+1)\{ j} 

  j,α f  j bζ g ∩ ν j ≤ α for every α ∈ Dg ∩ C j ∩ S j . sup ζ <μ

Proof of the claim Suppose otherwise.  Let H be the set of all α ∈ C j ∩ S j such that

  q  j,α bζ α ∩ ν j > α, where qα = f g . Then clearly H ∈ N Sν+j . ζ <μ 

  q  For α ∈ H , let πα = the least π < μ such that bζ α ∩ ν j \α = ∅, δα = ζ ≤π  q j the least δ < μ such that {ek(α j,θ) : θ < μ and k( j, θ ) ≤ π } ⊆ dα ∩ δ, and βα = μ ⊆ α and sup

j

the least β < α such that dα ∩ δα = z βα . Then we may find π < μ, δ < and K ∈ N Sν+j ∩ P(H ) such that for any α ∈ K , πα = π, δa = δ and βα   q q it is simple to see that bζ α = b γ for any two members α, γ ζ ≤π ζ ≤π ζ contradiction completes the proof of the claim.  Define M for  ≤ n + 1 by M = (Ci ∩ Si ) if  ≤ n, and ≤i≤n

μ, β < ν j = β. Now of K . This M = {∅}

N Sν+−1

∩ P(C−1 ∩ S−1 ) otherwise. By induction on , we will define u  : M −→  ≤ n + 1 such that the following holds. and a function v of domain M for 1 ≤  Let 2 ≤  ≤ n + 1. Then ran(v ) ⊆ (Cr ∩ Sr ), and given g ∈ M and r ≤−2 

  f α ∈ u  (g), sup bζ ∩ νr ≤ (v (g))(r ) for every r ≤  − 2, where f = ζ <μ

g ∪ {( − 1, α)} ∪ v (g). Let u 1 and v1 be the two functions defined on M1 by u 1 (g) = C0 ∩ S0 and v1 (g) = {∅}. Now suppose that 1 ≤  ≤ n, and u  and v have already been constructed. We define u +1 and v+1 as follows. Let g ∈ M+1 . that for any By the claim, there is m : C ∩ S −→ C−1 ∩ S−1 with the property   f bζ ∩ ν−1 ≤ m(β), where β ∈ C ∩ S , m(β) ∈ u  (g ∪ {(, β)}) and sup ζ <μ

f = g ∪ {(, β), ( − 1, m(β))} ∪ v (g ∪ {(, β)}). Then we may find ϕ, h and T such that T ∈ N Sν+ ∩ P(C ∩ S ), and m(β) = ϕ   and v g ∪ {(, β)} = h for every β ∈ T . Now set u +1 (g) = T and v+1 (g) = h ∪ {( − 1, ϕ)}. α)} for every α ∈ W . Put W = u n+1 (∅), ψ = vn+1 (∅), and f α = ψ ∪

{(n,  f  bζ α ∩ νn ≤ α. Put Applying the claim again, we get α ∈ W such that sup ζ <μ   fα f  a= eζ ∪ bζ α . Then clearly a ∈ A. Moreover, sup(a ∩ νn ) = α and for any ζ <μ

r ≤ n − 1, sup(a ∩ νr ) = ψ(r ). Thus sup(a ∩ νi ) ∈ Si for every i ≤ n.



Let us now discuss the condition in the statement of Proposition 6.1 that each Si lies in I [νi ]. We start by observing the following.

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  Proposition 6.2 Let S ∈ N Sκ+ ∩ P E μκ . Then {a ∈ Pκ (λ) : sup(a ∩κ) ∈ S} ∈ (N μSκ,λ )+ . Proof Let C be a μ-closed, cofinal subset of Pκ (λ). Let A denote the set of all infinite limit ordinals less than κ. By induction construct ai ∈ Pκ (λ) for i < κ such that • ai ⊂ ai+1 . • ai+1 ∈ C. • ai = j
be i ∈ A such that sup(ai ∩ κ) ∈ S. Then clearly, cf(i) = μ and ai ∈ C. μ

For A ⊆ κ, let K κ (A) denote the following two-person game of length μ. At step α < μ, I selects ξα ∈ κ, and II replies by playing ζα ∈ κ. The players must follow the rule that ζβ < ξα < ζα whenever β < α < μ. I wins just in case sup({ξα : α < μ}) ∈ A. μ It is simple to see that for A ⊆ κ, I has a winning strategy in K κ (A) just in case μ ∗ A ∈ (N G κ,κ ) . More generally, the following holds. Proposition 6.3 Let ν be a regular cardinal with κ ≤ ν ≤ λ, and A ⊆ ν. Then the following are equivalent : μ

(i) I has a winning strategy in K ν (A). μ ∗ (ii) {a ∈ Pκ (λ) : sup(a ∩ ν) ∈ A} ∈ (N G κ,λ ) . μ

Proof (i)→(ii) : Let σ be a winning strategy for I in K ν (A). Set B = {a  ∈ Pκ (λ) : μ sup(a ∩ ν) ∈ A}. We define a winning strategy τ for II in Hκ,λ B, Pκ (λ) as follows. Consider a run of the game where I successively plays a0 , a1 , . . .. Define ξi and ζi for i < μ by • ξ0 = σ (∅). • ζi = the least η ∈ν such that η > ξi and η ≥ sup(ai ∩ ν). • For i > 0, ξi = σ ζ j : j < i . Now put τ (a0 , . . . , ai ) = {ξi+1 }. μ (ii)→(i) : Given a winning strategy τ for II in G κ,λ (B), where B = {a ∈ Pκ (λ) : μ sup(a ∩ ν) ∈ A}, we define a winning strategy σ for I in K ν (A) as follows. Let ζi ∈ ν for i < μ. We define ξi ∈ ν, ai ∈ Pκ (λ) and bi ∈ Pκ (λ) for i < ν by • • • •

ξ0 = 0 and a0 = {ζ0 + 1}. bi = τ (a0 , . . . , ai ). ai+1 = bi ∪ {ζi+1 + 1} and ξi+1 = sup(bi+1 ∩ ν). ai = a j and ξi = sup({ξ j : j < i}) in case i is an infinite limit ordinal. j
Now suppose that ξi < ζi for all i < μ. Then we let σ (∅) = ξ0 and for each 

i ∈ μ\{0}, σ (ζ j : j < i) = ξi . Thus if ν is a regular cardinal with κ ≤ ν ≤ λ, and T ∈ N Sν+ ∩ P(E μν ) is such that μ μ I has a winning strategy in K ν (ν\T ), then {a ∈ Pκ (λ) : sup(a ∩ ν) ∈ T } ∈ N G κ,λ . For instance, suppose that in V, GCH holds and χ , ρ and θ are three cardinals such that ω < χ = cf(χ ) < ρ ≤ cf(θ ) and ρ is Mahlo. Then by a result of [16], there is a generic extension W of V in which

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• Cardinals ≤ χ and cardinals ≥ ρ are preserved. • χ + = ρ and 2χ = θ . χ+ χ • There is T ∈ N Sχ++ ∩ P(E χ ) such that I has a winning strategy in K χ + (χ + \T ). It follows from Propositions 6.2 and 6.3 that in W , χ

{a ∈ Pχ + (σ ) : sup(a ∩ χ + ) ∈ T } ∈ N G χ + ,σ \N χ -Sχ + ,σ for every cardinal σ > χ . Lemma 6.4 (Donder and Levinski [5], Huuskonen et al. [15]) Suppose κ <μ = κ. μ Then I has a winning strategy in K κ (A) for some A ∈ I [κ]. By a result of Mitchell [31], it is consistent relative to a large cardinal that no stationary subset of E ωω12 is in I [ω2 ]. Thus the conclusion of Lemma 6.4 may fail if we only assume that cov(κ, μ, μ, 2) = κ. Note that it immediately follows from Lemma 6.4 that if κ <μ = κ, then I [κ] ∩ N Sκ+ ∩ P(E μκ ) = ∅. / I [ν], where ν is a regular cardinal with κ ≤ ν = Proposition 6.5 Suppose that E μν ∈ ν <μ ≤ λ. Then there is T ∈ N Sν+ ∩ P(E μν ) such that {a ∈ Pκ (λ) : sup(a ∩ ν) ∈ T } ∈ μ N G κ,λ . Proof By Proposition 6.3 and Lemma 6.4.



For instance, let ρ be a supercompact cardinal, and σ be a cardinal with cf(σ ) < ρ < σ . Then by a result of Shelah (see Theorem 3.20 in [7]), we may find a singular + cardinal θ < ρ such that cf(θ ) = cf(σ ) and E θσ+ ∈ / I [σ + ]. Now suppose that μ = θ + μ + and κ ≤ σ ≤ λ. Then by Proposition 6.5, {a ∈ Pκ (λ) : sup(a ∩ σ + ) ∈ T } ∈ N G κ,λ   + for some T ∈ N Sσ++ ∩ P E μσ . We next show that there are situations when κ < ν = λ < ν <μ and the conclusion of Proposition 6.5 holds.  Proposition 6.6 Let π > λ be a regular

 cardinal. Suppose that f =  f α : α < π  is an increasing, cofinal sequence in λi , < I , where i<ν

• ν is a cardinal with cf(λ) ≤ ν < κ. • λi : i < ν is a one-to-one sequence of regular cardinals less than λ such that ν < λ0 and sup({λi : i < ν}) = λ. • I is a proper ideal on ν such that for any cardinal χ < λ, {i ∈ ν : λi ≤ χ } ∈ I . μ ∗ Then the set of all x ∈ Pκ (π ) such that sup(x) is a good point for f lies in (N G κ,π ) .

Proof Letting X = {x ∈ Pκ (π ) : sup(x) is a good point for f}, we define a winning μ μ strategy τ for II in G κ,π (X ) as follows. Consider a run of the game  G κ,π (X ), where λi by gγ (i) = I’s successive moves are z 0 , z 1 , . . .. Given γ < μ, define gγ ∈ i∈ν sup({ f α (i) : α ∈ z γ ∪ {sup(z γ )}}) if λi ≥ κ, and gγ (i) = 0 otherwise. Pick βγ ∈ π so that gγ < I f βγ . Note that sup(z γ ) < βγ . We put τ (z 0 , . . . , z γ ) = z γ ∪ {βγ }.

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  Let us verify that z γ ∈ X . Set δ = sup z γ and for every γ < γ <μ γ <μ   μ, Z γ = i ∈ ν : gγ (i) ≥ f βγ (i) ∪ {i ∈ ν : λi < κ}. It is immediate that {βγ : γ < μ} is a cofinal subset of δ. Now let η < γ < μ. Then βη ∈ z γ , so 

f βη (i) ≤ gγ (i) < f βγ (i) for all i ∈ ν\Z γ . Corollary 6.7 (Cummings et al. [3]) Suppose that f =  f α : α < π  is as in the statement of Proposition 6.6, and S ∈ I [π ] ∩ P(E μπ ). Then the set of all γ ∈ S such that γ is not a good point for f lies in N Sπ . 

Proof By Propositions 6.1 and 6.6.

Corollary 6.8 Let χ and π be two cardinals such that cf(λ) ≤ χ < κ, λ < π ≤ ppχ (λ), and π is not the successor of a singular cardinal. Then there is an Aκ,λ (χ + , π )sequence yα : α < π  such that |{α < π : yα ⊆ a}| = |a| for all a ∈ Pχ (λ). Proof Case when π is a successor cardinal : By Lemmas 3.3.5 and 9.1.1 of [14], we may find ν, λi : i < ν, I and f =  f α : α < π  such that • ν is a cardinal with cf(λ) ≤ ν ≤ χ . • λi : i < ν is a one-to-one sequence of regular cardinals less than λ such that ν < λ0 and sup({λi : i < ν}) = λ. • I is a proper ideal on ν such that for any cardinal σ < λ, {i ∈ ν : λi ≤ σ } ∈ I .   • f is an increasing, cofinal sequence in λi , < I . i<ν

By Proposition 5.7 (iii) and Corollary 6.7 there is a closed unbounded subset C of π such that every limit ordinal δ in C with cf(δ) < λ is a good point for f. Let cα : α < π  be the increasing enumeration of C. Pick a bijection ϕ : ν × λ → λ. We define yα for α < π by : yα = {α} if α < λ, and yα = {ϕ(i, f cα (i) : i < ν} otherwise. Then by Proposition 4.16 (ii) of [22], the sequence yα : α < π  is as desired. Case when π is a singular cardinal : Use Proposition 3.3 of [22]. 

Corollary 6.9 Suppose that f =  f α : α < π  is as in the statement of Proposition 6.5, and S ∈ N Sπ+ ∩ P(E μπ ) is such that no γ ∈ S is a good point for f. Then μ {x ∈ Pκ (π ) : sup(x) ∈ S} ∈ N G κ,π . Proof By Proposition 6.6.



Starting from a supercompact cardinal, Gitik and Sharon [12] have constructed a model M of ZFC in which there is a strong limit cardinal σ of cofinality ω with the following properties : • 2σ > σ + . • σ carries a very good scale of length σ + and a very good scale of length σ ++ . • (as observed by Cummings and Foreman) σ carries a scale f of length σ + that is not good. Thus in M, we may find a regular cardinal μ with ω < μ < σ , and S ∈ N Sσ++ ∩ + P(E μσ ) such that no γ ∈ S is a good point for f. Set λ = σ + . Then (λ )<μ > λ and by Corollary 6.9, for any regular cardinal κ with μ < κ < σ, {a ∈ Pκ (λ ) : μ sup(a) ∈ S} ∈ N G κ ,λ . We conclude this section with the following variant of Proposition 6.1.

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Proposition 6.10 Let ρ and ν be two cardinals such that κ ≤ ρ < ν = cf(ν) ≤ λ and μ + (cov(ρ, κ, κ, μ+ ))<μ < ν. Further let T ∈ (N G κ,ρ ) and S ∈ N Sν+ ∩ I [ν] ∩ P(E μν ). μ

+

Then {a ∈ Pκ (λ) : a ∩ ρ ∈ T and sup(a ∩ ν) ∈ S} ∈ (N G κ,λ ) . μ

Proof Pick B ∈ Z κ,ρ with |B| = cov(ρ, κ, κ, μ+ ). We may find C, z β : β < ν and dα : α ∈ C ∩ S such that (a) C is a closed unbounded subset of ν, and (b) for each α ∈ C ∩ S, dα is an unbounded subset of α of order-type μ with the property that {dα ∩ δ : δ < α} ⊆ {z β : β < α}. For α ∈ C ∩ S, let ξθα : θ < μ be  μ ∗ the increasing enumeration of dα . Now fix A ∈ N G κ,λ . Pick a winning strategy μ τ for player II in Hκ,λ (A, Pκ (λ)). Let X be the set of all α ∈ C ∩ S such that ν ∩ τ (k(0) ∪ {ξ0α }, . . . , k( j) ∪ {ξ αj }) ⊆ α whenever j < μ and k : j + 1 → B. Then it is simple to see that (C ∩ S)\X ∈ N Sν . Pick α ∈ C ∩ S ∩ X . Let W be the set of all x ∈ Pκ (ρ) for which one can find f x : μ → B such that x=



  f x ( j) ∪ ρ ∩ τ ( f x (0) ∪ {ξ0α }, . . . , f x ( j) ∪ {ξ αj }) .

j<μ μ

Clearly, II has a winning strategy in K κ,ρ (W, B), so by Proposition 2.13 W ∈ μ ∗ (N G κ,ρ ) . Pick x ∈ W ∩ T and set x=



  f x ( j) ∪ {ξ αj } ∪ τ f x (0) ∪ {ξ0α }, . . . , f x ( j) ∪ {ξ αj } .

j<μ

Then clearly, a ∈ A. Moreover, a ∩ ρ = x and sup(a ∩ ν) = α.



In particular, if ρ and ν are two cardinals such that κ ≤ ρ and cov(ρ, κ, κ, ω1 ) < ν = cf(ν) ≤ λ, then {a ∈ Pκ (λ) : a ∩ ρ ∈ T and sup(a ∩ ν) ∈ S} ∈ (N G ωκ,λ )+ for every T ∈ (N G ωκ,ρ )+ and every S ∈ N Sν+ ∩ P(E ων ). 7 Applications of scales  Suppose that λ < π = tcf( A/I ),  where A is an infinite set of regular cardinals with |A| < min(A) and cf sup(A) ≤ |A| < κ < sup(A) ≤ λ, and I is an ideal on A such that {A ∩ a : a ∈ A} ⊆ I . Let f =  f ξ : ξ < π  be an increasing, cofinal sequence in (A, < I ). Define h : Pκ (λ) → π by h(b) = the least ξ such that {a : sup(b ∩ a) ≤ f ξ (a)} ∈ I ∗ . μ We will see that there are situations in which h(N G κ,λ ) is a restriction of N Sπ . Proposition 7.1 Iπ ⊆ h(Iκ,λ ). Proof Clearly, h −1 (ξ ) ∩ {b ∈ Pκ (λ) : ran( f ξ ) ⊆ b} = ∅ for all ξ < π . The result follows. 

∗ . Lemma 7.2 {x ∈ Pκ (π ) : h(x ∩ λ) ≥ sup(x)} ∈ N Sκ,π

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Proof Simply observe that {x ∈ Pκ (π ) : ∀ξ ∈ x (ran( f ξ ) ⊆ x)} ⊆ {x ∈ Pκ (π ) : h(x ∩ λ) ≥ sup(x)}. 

Lemma 7.3 Suppose that either I is μ+ -complete, or μ > cof(I ). Let b j : j < μ be   an increasing sequence in (Pκ (λ), ⊆). Then h b j = sup({h(b j ) : j < μ}). j<μ

Proof Set ξ = sup({h(b j ) : j < μ}). It is immediate that h

 j<μ

 b j ≥ ξ . Let us

prove the reverse inequality. For j < μ, set ξ j = h(b j ), Q j = {a : sup(b j ∩ a) ≤ f ξ j (a)} and R j = {a : f ξ j (a) ≤ f ξ (a)}. (Q j ∩ R j ) ∈ I ∗ . Clearly, First assume that I is μ+ -complete. Then j<μ    (b j ∩ a) ≤ sup({ f ξ j (a) : j < μ}) ≤ f ξ (a) for every a ∈ (Q j ∩ sup j<μ j<μ   bj ≤ ξ. R j ), so h j<μ

Let us now deal with the case when cof(I ) < μ. Select F ⊆ I ∗ with |F| < μ so that for any H ∈ I ∗ , there is K ∈ F with K ⊆ H . Define ϕ : μ −→ F so that ϕ( j) ⊆ Q j ∩ R j for all j < μ. We may find K ∈ F and x ⊆ μ with |x| = μ such that ϕ( j) = K for every j ∈ μ. Then sup



   (b j ∩ a) = sup (b j ∩ a) ≤ sup({ f ξ j (a) : j ∈ x}) ≤ f ξ (a)

j<μ

for each a ∈ K , and therefore h

j∈x



 bj ≤ ξ.



j<μ

Proposition 7.4 Suppose that either I is μ+ -complete, or μ > cof(I ). Then {x ∈ μ Pκ (π ) : h(x ∩ λ) = sup(x)} ∈ (N G κ,π )∗ . μ

Proof By Lemma 7.2, it suffices to show that X ∈ (N G κ,π )∗ , where X = {x ∈ μ Pκ (π ) : sup(x) ≥ h(x ∩ λ)}. We define a winning strategy τ for II in G κ,π (X ) as follows. Consider a run of the game where I’s successive  moves are x0 , x1 , . . .. We let τ (x0 , . . . , xα ) = xα ∪ {h(xα ∩ λ)}. We do have that xα ∈ X , since by α<μ 

  

  Lemma 7.3, h xα ∩ λ = sup({h(xα ∩ λ) : α < μ}) ≤ sup xα . α<μ α<μ 

Corollary 7.5 Suppose that either I is μ+ -complete, or μ > cof(I ). Then N Sπ |E μπ ⊆ μ h(N G κ,λ ) ⊆ N Sπ |S for every S ∈ N Sπ+ ∩ P(E μπ ) ∩ I [π ]. μ

Proof To show that N Sπ | E μπ ⊆ h(N G κ,λ ), let C be a closed unbounded subset of π . Then by Proposition 7.4, ∗ {x ∈ Pκ (π ) : h(x ∩ λ) = sup(x) ∈ C ∩ E μπ } ∈ (N G μ κ,π ) ,

  μ and consequently by Proposition 2.14, h −1 C ∩ E μπ ∈ (N G κ,λ )∗ .

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Ideals on Pκ (λ) associated with games of uncountable length μ

Now let S ∈ N Sπ+ ∩ P(E μπ ) ∩ I [π ]. We will show that h(N G κ,λ ) ⊆ N Sπ | S. Thus fix W ∈ N Sπ+ ∩ P(S) ∩ P(π \μ). Then by Propositions 6.1 and 7.4, {x ∈ μ Pκ (π ) : h(x ∩ λ) = sup(x) ∈ W } ∈ (N G κ,π )+ and therefore by Proposition 2.14, μ + −1 

h (W ) ∈ (N G κ,λ ) . Corollary 7.6 Suppose that E μπ ∈ I [π ] and either I is μ+ -complete, or μ > cof(I ). μ Then h(N G κ,λ ) = N Sπ | E μπ . Proposition 7.7 Suppose that π <μ = π and either I is μ+ -complete, or μ > cof(I ). μ Then there is S ∈ N Sπ+ ∩ P(E μπ ) such that h(N G κ,λ ) = N Sπ | S. μ

Proof By Lemma 6.4 there is T ∈ I [π ] such that I has a winning strategy σ in K π (T ). Set S = T ∩ E μπ . μ

∗

Claim Let C be a closed unbounded subset of π . Then h −1 (C ∩ S) ∈ (N G κ,λ ) .  μ  Proof of the claim We define a winning strategy τ for II in G κ,λ h −1 (C ∩ S) as follows. Consider a run of the game where I successively plays b0 , b1 , · · · . Define ξα and ζα for α < μ by • ξ0 = σ (∅).   • ζα = the least η ∈ C with η ≥ max sup({h(bβ ) : β ≤ α}), ξα + 1 . • For α > 0, ξα = σ (ζβ : β < α).   Now let τ (b0 , . . . , bα ) = bα ∪ f ζα (a) + 1 : a ∈ A . Put ζ = sup({ζα : α < μ}). It is simple to see that S. Furthermore, by Lemma 7.3, ζ = sup({h(bα ) : α <  ζ ∈ C ∩

 bα . Hence bα ∈ h −1 (C ∩ S), which completes the proof of μ}) = h α<μ α<μ the claim. We can infer from the claim that S ∈ N Sπ+ ∩ P(E μπ ), and also that N Sπ | S ⊆ μ

 h(N G κ,λ ). The reverse inclusion holds by Corollary 7.5. Corollary 7.8 Suppose that cf(λ) ∈ κ\{μ} and λ<μ ≤ λ+ . Then there is ρ : + μ + + λ Pκ (λ) −→ λ and S ∈ N Sλ+ ∩ P E μ such that ρ(N G κ,λ ) = N Sλ+ | S (and μ hence cof(N Sλ+ ) ≤ cof(N G κ,λ )). 

Proof By Propositions 5.1 and 7.7.

Corollary 7.9 (i) Suppose that cf(λ) ∈ κ\{μ} and 2λ = λ+ . Then there is no A μ such that N G κ,λ = N Sκ,λ | A. (ii) Suppose that μ < cf(λ) < κ, λ<μ = λ, 2λ = λ+ and there exists a [λ]<μ μ [λ]<μ normal ideal on Pκ (λ). Then there is no A such that N G κ,λ = N Sκ,λ | A. μ

(i) By Corollary 7.8, cof(N Sκ,λ ) ≤ 2λ = λ+ < cof(N Sλ+ ) ≤ cof(N G κ,λ ). The desired conclusion easily follows. <μ [λ]<μ ) ≤ λ(λ ) = 2λ . Hence, as in the proof of (i), (ii) By Proposition 1.2, cof(N Sκ,λ

Proof

<μ

μ

[λ] cof(N Sκ,λ ) < cof(N G κ,λ ).



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Let us now consider situations when the conclusion of Proposition 7.4 fails. The following generalizes a result of Zapletal (see Example 35 in [10]). Proposition 7.10 Suppose that cf(sup(A)) = |A| = μ and (sup(A))<μ < π , and let  : π → π and S ∈ I [π ] ∩ N Sπ+ ∩ P(E μπ ). Then {x ∈ Pκ (π ) : sup(x) ∈ S and  μ + h(x ∩ λ) > (sup(x))} ∈ N G κ,π . Proof Let ai : i < μ be a one-to-one enumeration of A. We may find a closed unbounded subset C of π, z β : β < π  and dα : α ∈ C ∩ S such that for any α ∈ C ∩ S, dα is a cofinal subset of α of order-type μ and {dα ∩ δ : δ < α} ⊆ {z β : β < α}. For α ∈ C ∩ S, let ξθα : θ < μ be the increasing enumeration of dα . Now  μ ∗ μ fix X ∈ N G κ,π . Let τ be a winning strategy for II in G κ,π (X ). For α ∈ C ∩ S, define xiα and tiα for i < μ by • x0α = {ξ0α } ∪ { f (α) (a0 ) + 1}. • tiα = τ (x0α , . . . , xiα ).  • For i > 0, xiα = {ξiα } ∪

tα j


∪ { f (α) (ai ) + 1}.

Let Q be the set of all α in C ∩ S such that sup(xiα ) ≥ α for some i < μ. Claim Q ∈ N Sπ . Proof of the claim Suppose otherwise. There must be R ∈ N Sπ+ ∩ P(Q) and j < μ such that sup(x αj ) ≥ α for all α ∈ R. Now we may find T ∈ N Sπ+ ∩ P(R), q ∈  ai and β < π such that for each α ∈ T , {ξiα : i ≤ j} = z β and  f (a) (ai ) : i≤ j

γ

i ≤ j = q(i) : i ≤ j. But then x αj = x j whenever α, γ ∈ T . This contradiction completes the proof of the claim.  Pick α in (C ∩ S)\Q, and put x = xiα . Then clearly, x ∈ X and sup(x) = α. i<μ

Moreover, { f (α) (a) + 1 : a ∈ A} ⊆ x, and consequently h(x ∩ λ) > (α).



On the other hand one can prove the following which generalizes Lemma 2.5 in [11]. Proposition 7.11 Suppose that cof(I ) < κ, and χ <μ < κ for every cardinal χ < κ. Suppose further that there is a closed unbounded subset C of π with the property that every δ in C ∩ E κπ is a remarkably good point for f. Then for each S in N Sκ+ ∩  μ + P(E μκ ), {x ∈ Pκ (π ) : x ∩ κ ∈ S and h(x ∩ λ) = sup(x)} ∈ N G κ,π . Proof Fix S ∈ N Sκ+ ∩ P(E μκ ). By Lemma 7.2, it suffices to show that {x ∈ Pκ (π ) :   μ + μ ∗ x ∩ κ ∈ S and h(x ∩ λ) ≤ sup(x)} ∈ N G κ,π . Thus fix X ∈ N G κ,π , and let μ τ be a winning strategy for II in Hκ,π (X, Pκ (π )). For β < κ, set rβ = {γ < μ : γ ≤ β}. Given β < κ and γ ∈ rβ , let K γβ denote the set of all increasing functions k : γ + 1 → β + 1. Note that |K γβ | < κ. We define eβ ∈ Pκ (π ), dβ ∈ Pκ (π ) and ζβ ∈ C\λ for β < κ as follows: • ζ0 = min(C\λ) and e0 = {ζ0 }.

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Ideals on Pκ (λ) associated with games of uncountable length

 • If β is an infinite limit ordinal, then eβ = eγ and ζβ = sup({ζγ : γ < β}). γ <β     K γβ }, where z k = • dβ = eβ ∪ (sup(eβ ∩ κ)) + 1 ∪ {z k : k ∈ γ ∈rβ   τ ek(0) , . . . , ek(γ ) if γ ∈ rβ and k ∈ K γβ . • ζβ+1 = the least ζ in C\(ζ + 1) such that sup(dβ ) ≤ ζ and h(dβ ∩ λ) ≤ ζ . • eβ+1 = dβ ∪ {ζβ+1 }. Note that the sequence ζβ : β < κ is increasing. Moreover for any β < κ, sup(eβ ) = ζβ and h(eβ+1 ∩ λ) ≤ ζβ+1 . Since sup({ζβ : β < κ}) is a remarkably good point for f, we may find a closed unbounded subset D of κ, and ϕ : D → I / ϕ(β) ∪ ϕ(η). Define such that f ζβ (a) < f ζη (a) whenever β < η are in D and a ∈ s : D → D by s(β) = min(D\(β + 1)), and pick ψ : D → I so that for any β in D, • ϕ(s(β)) ⊆ ψ(β), and • sup(eβ+1 ∩ a) ≤ f ζs(β) (a) if a ∈ A\ψ(β). We may find a stationary subset T of D, and Z ∈ I such that ψ takes the constant value Z on T . Now pick β in S so that T ∩ β is cofinal in β and {sup(eγ ∩ κ) : γ < β} ⊆ β. Let βi : i < μ be anincreasing sequence of elements of T with supremum β. Then clearly, τ eβ0 , . . . , eβi ⊆ dβi ⊆ eβi +1 ⊆ eβi+1 for every i < μ, and  therefore  eβ ∈ X . Moreover, eβ ∩κ = β. Given a in A\(Z ∪ϕ(β)), we have that sup eβi ∩a ≤ f ζs(βi ) (a) < f ζβ (a) for all i < μ, and consequently sup(eβ ∩ a) ≤ f ζβ (a). It follows 

that h(eβ ∩ λ) ≤ ζβ = sup(eβ ). Let us observe that a much better result (see [23]) holds in the case when μ = ω. 8 More applications of scales μ

μ

In this section we consider situations when f (N G κ,λ | W ) ⊆ N G κ,π | X for some μ

+

regular cardinal π > λ, f : Pκ (λ) −→ Pκ (π ), W ∈ (N G κ,λ )+ and X ∈ (N G κ,π ) . Throughout the section π will denote a regular cardinal greater than λ. Let p : Pκ (π ) −→ Pκ (λ) be defined by p(x) = x ∩ λ. Given an Aκ,λ (κ, π )-sequence y = yα : α < π , we set D y = {x ∈ Pκ (π ) : ∀α ∈ x (yα ⊆ x)}, A(y ) = {x ∈ Pκ (π ) : {α < π : yα ⊆ x} ⊆ x}, W y = p“ D y ∩ A(y ) and R y = p“({x ∈ D y ∩ A(y ) : |x| = |x ∩ κ|}), and we define f y : Pκ (λ) −→ Pκ (π ) by f y (a) = {α < π : yα ⊆ a}. Note that R y ⊆ W y . The following is readily checked.

Lemma 8.1 Let y = yα : α < π  be an Aκ,λ (κ, π )-sequence. Then the following hold: ∗ . (i) D y ∈ N Sκ,π   (ii) Let a ∈ Pκ (λ). Then p f y (a) = a. (iii) Let x ∈ D y ∩ A(y ). Then f y p(x) = x.

Lemma 8.2 Let y = yα : α < π  be an Aκ,λ (κ, π )-sequence, and let Y ⊆ D y ∩ A(y ). Then p“ Y = f y−1 (Y ).

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  Proof Set B = p“ Y . If a ∈ f y−1 (Y ), then by Lemma 8.1 p f y (a) = a, and consequently a ∈ B. Thus f y−1 (Y ) ⊆ B. To prove the reverse inclusion, fix b ∈ B.

Select x ∈ Y so that p(x) = b. Now by Lemma 8.1 f y (b) = x, so b ∈ f y−1 (Y ).



Proposition 8.3 Let y = yα : α < π  be an Aκ,λ (κ, π )-sequence. Then the following hold : (i) Let a ∈ R y . Then | f y (a)| = |a ∩ κ|. μ + μ + (ii) If A(y ) ∈ (N G κ,π ) , then R y ∈ (N G κ,λ ) . μ ∗ μ ∗ (iii) If A(y ) ∈ (N G κ,π ) , then R y ∈ (N G κ,λ ) . (i) By Lemma 8.2, R y = f y−1 ({x ∈ D y ∩ A(y ) : |x| = |x ∩ κ|}). Hence for any a ∈ R y , | f y (a)| = |κ ∩ f y (a)| = |a ∩ κ|.  μ + / (ii) Suppose that A(y ) ∈ N G κ,π . Then by Propositions 2.3 and 2.5, p −1 (R y ) ∈ μ N G κ,π , since A(y ) ∩ D y ∩ {x ∈ Pκ (π ) : |x| = |x ∩ κ|} ⊆ p −1 (R y ). Hence by μ / N G κ,λ . Proposition 2.14, R y ∈  μ ∗ (iii) Suppose that A(y ) ∈ N G κ,π . Set B = Pκ (λ)\R y . Then by Propositions 2.3 μ and 2.5, p −1 (B) ∈ N G κ,π , since p −1 (B) ∩ (D y ∩ A(y ) ∩ {x ∈ Pκ (π ) : |x| = μ 

|x ∩ κ|}) = ∅, and therefore by Proposition 2.14 B ∈ N G κ,λ .

Proof

Proposition 8.4 Let y = yα : α < π  be an Aκ,λ (κ, π )-sequence such that A(y ) ∈ μ + (N G κ,π ) . Then the following hold: (i) f y (Iκ,λ | W y ) = Iκ,π | (D y ∩ A(y )). μ μ (ii) f y (N G κ,λ | R y ) ⊆ N G κ,π | A(y ). μ ∗ μ μ (iii) If A(y ) ∈ (N G κ,π ) , then f y (N G κ,λ ) = N G κ,π .   Proof Let us first show that Iκ,π | D y ∩ A(y ) ⊆ f y (Iκ,λ | W y ). Thus fix Q ∈ Iκ,π |  D y ∩ A(y ) . Pick t ∈ Pκ (π ) so that {x ∈ Q ∩ D y ∩ A(y ) : t ⊆ x} = ∅.  Claim 1 {v ∈ f y−1 (Q) ∩ W y : yα ⊆ v} = ∅. α∈t

Proof of Claim 1 Suppose otherwise, and select v ∈ f y−1 (Q)∩ W y so that



yα ⊆ v.

α∈t

Then clearly, t ⊆ f y (v) ∈ Q. Moreover by Lemma 8.2, f y (v) ∈ D y ∩ A(y ). This contradiction completes the proof of Claim 1. It immediately follows from Claim 1 that Q ∈ f y (Iκ,λ | Wy ).  Next we show that conversely, f y (Iκ,λ | W y ) ⊆ Iκ,π | D y ∩ A(y ) . Thus let K ∈ f y (Iκ,λ | W y ). Pick c ∈ Pκ (λ) so that {a ∈ f y−1 (K ) ∩ W y : c ⊆ a} = ∅. Claim 2 {z ∈ K ∩ D y ∩ A(y ) : c ⊆ z} = ∅. Proof of Claim 2 Suppose otherwise, and pick z ∈ K ∩ D y ∩ A(y ) with c ⊆ z. Then clearly, c ⊆ p(z) ∈ W y . Moreover, p(z) ∈ f y−1 (K ), since by Lemma 8.1   f y p(z) = z. This contradiction completes the proof of Claim 2.

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Ideals on Pκ (λ) associated with games of uncountable length

  It is an immediate consequence of Claim 2 that K ∈ Iκ,π | D y ∩ A(y ) .  + μ Let us now show that (ii) holds. Suppose X ∈ N G κ,π | A(y ) . Put Z = {x ∈ X ∩ D y ∩ A(y ) : |x| = |x ∩ κ|} and B = p“ Z . Note that B ⊆ R y . Furμ + thermore Z ⊆ p −1 (B), so by Proposition 2.14 B ∈ (N G κ,λ ) . It follows that +

μ

R y ∩ f y−1 (X ) ∈ (N G κ,λ ) , since by Lemma 8.2 B = f y−1 (Z ) ⊆ R y ∩ f y−1 (X ). μ Hence X ∈ / f y (N G κ,λ | R y ). μ ∗ It remains to prove (iii). Thus suppose that A(y ) ∈ (N G κ,π ) . Then by (ii) and μ μ Proposition 8.3, f y (N G κ,λ ) ⊆ N G κ,π . To show the reverse inclusion, fix T ∈    μ + μ + f y (N G κ,λ ) . Set H = D y ∩ A(y ) ∩ p −1 f y−1 (T ) . Then H ∈ (N G κ,π ) since     μ by Proposition 2.14 p −1 f y−1 (T ) ∈ / N G κ,π . Now by Lemma 8.1, f y p(x) = x for +

μ

every x ∈ H , so H ⊆ T , and hence T ∈ (N G κ,π ) . μ



∗

μ

Our next observation is that if A(y ) ∈ (N G κ,π ) , then N G κ,λ satisfies a strong normality condition. Let J be an ideal on Pκ (λ), and y = yα : α < π  be a (κ, λ, π )-sequence. J is y-normal if for any H ∈ J + , and any g : H −→ {yα : α < π } with the property that g(a) ⊆ a for every a ∈ H , there is B ∈ J + ∩ P(H ) such that g is constant on B. Note that if J is y-normal, then it is normal. Proposition 8.5 Let y = yα : α < π  be an Aκ,λ (κ, π )-sequence such that A(y ) ∈ μ ∗ μ (N G κ,π ) . Then N G κ,λ is y-normal. μ

+

Proof Fix H ∈ (N G κ,λ ) and h : H −→ π with the property that yh(a) ⊆ a for every μ

+

a ∈ H . Set X = D y ∩ A(y ) ∩ p −1 (H ). Then by Proposition 2.14, X ∈ (N G κ,π ) . Define k : X −→ π by k(x) = h( p(x)). For any x ∈ X, yk(x) ⊆ p(x), so by μ + Lemma 8.1 k(x) ∈ x. Hence we may find Y ∈ (N G κ,π ) ∩ P(X ) and β ∈ π such that k takes the constant value β on Y . Set B = p“ Y . Then clearly, B ⊆ H . Since μ + Y ⊆ p −1 (B), we can infer from Proposition 2.14 that B ∈ (N G κ,λ ) . It is easy to check that h takes the constant value β on B. 

μ

+

It remains to see whether we can find y such that A(y ) lies in (N G κ,π ) (or even μ ∗ in (N G κ,π ) ). Proposition 8.6 Suppose that y = yα : α < π  is an Aκ,λ (μ, π )-sequence. Then A(y ) is μ-club.  μ  Proof We define a winning strategy τ for II in G κ,π A(y ) by τ (a0 , . . . , aα ) = aα ∪ μ ∗ {β < π : yβ ⊆ aα }. As A(y ) lies in (N G κ,π ) , it is a cofinal subset of Pκ (π ). One easily verifies that A(y ) is μ-closed. 

I (ν + , π )-sequence, where Proposition 8.7 Suppose that y = yα : α < π  is a Dκ,λ ν is an infinite cardinal less than κ, and I a proper, μ+ -complete ideal on ν. Then A(y ) ∈ (N μ-Sκ,π )∗ .

Proof There must be a bijection ψ : ν × λ −→ λ, and f α : ν −→ λ for λ ≤ α < π such that

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• For any nonempty e in Pκ + (π \λ), there is g : e −→ I such that f α (i) = f β (i)  whenever α < β are in e and i ∈ ν\ g(α) ∪ g(β) .  • For λ ≤ α < π, yα = {ψ i, f α (i) : i < ν}.   Define h : (π \λ) × Pκ (λ) −→ P(ν) by h(α, a) = {i ∈ ν : ψ i, f α (i) ∈ a}. Let X be the set of all x ∈ Pκ (π ) such that {α ∈ π \λ : h(α, x ∩ λ) ∈ I + } ⊆ x. We μ define a winning strategy τ for II in G κ,π (X ) by τ (x0 , . . . , xβ ) = xβ ∪ {α ∈ π \λ : μ ∗ + h(α, xβ ∩ λ) ∈ I }. Thus X ∈ (N G κ,π ) . Since X is obviously μ-closed, it follows 

that X is μ-club. But clearly X ⊆ A(y ), so A(y ) ∈ (N μ-Sκ,π )∗ . Our next result shows that if κ = ω1 (and hence μ = ω), then there is no Aκ,λ (κ, π )μ ∗ sequence y such that A(y ) ∈ (N G κ,π ) . Proposition 8.8 Suppose that λ<μ < π and y = yα : α < π  is an Aκ,λ (μ+ , π )sequence. Let S ∈ N Sπ+ ∩ P(E μπ )∩ I [π ] and  : π → π . Then {x ∈ Pκ (π ) : sup(x) ∈  μ + S and y(sup(x)) ⊆ x} ∈ N G κ,π . Proof The proof is an easy modification of that of Proposition 7.10.



Proposition 8.9 Suppose that y = yα : α < π  is a Bκ,λ (κ, π )-sequence, and μ + S ∈ I [κ] ∩ N Sκ+ ∩ P(E μκ ). Then {x ∈ A(y ) : x ∩ κ ∈ S} ∈ (N G κ,π ) . Proof There must be a closed unbounded subset C of κ, z γ ∈ {w ⊆ κ : o.t.(w) < μ} for γ < κ, and ψ : C ∩ S → P(κ) such that for each β ∈ C ∩ S, ψ(β) is an unbounded subset of β of order-type μ and {ψ(β) ∩ δ : δ < β} ⊆ {z γ : γ < β}. For γ γ < κ, let ξk : k < o.t.(z γ ) be the increasing enumeration of z γ . For β < κ, let u β be the set of all γ ≤ β such that z γ ⊆ β and o.t.(z γ ) is a successor ordinal. Now  μ ∗ μ  fix X ∈ (N G κ,π ) , and let τ be a winning strategy for II in Hκ,π X, Pκ (π ) . Define eβ ∈ Pκ (π ) for β < κ by • e0 = ∅.  • eβ =

γ <β

eγ if β is an infinite limit ordinal.



 • eβ+1 = eβ ∪ ((sup(eβ ∩ κ)) + 1) ∪ {α < π : yα ⊆ eβ } ∪ yα ∪ α∈eβ

  τ (eξ γ : k < o.t.(z γ )) . k γ ∈u β   Set e = eβ , and select a < κ-to-one g ∈ yα . Let D be the set of all β<κ  −1α∈e β < κ such that {sup(eζ ∩ κ) : ζ < β} ⊆ β, and {g ({ξ }): ξ ∈ eη ∩ ran(g)} ⊆ eβ ∗ −1 for every η < β. Then  D ∈ N Sκ , since for any η < κ, {g ({ξ }) : ξ ∈ eη ∩ ran(g)} ∈ Pκ (e) = P(eβ ). Pick β ∈ D ∩ C ∩ S, and let βi : i < μ be β<κ  eβi , and moreover the increasing enumeration of ψ(β). Then clearly, eβ = i<μ

eβ ∩ κ = β. For j < μ, pick  j < μ and γ j ≤ β j so that {βi : i ≤ j} = z γ j ,   and put ζ j = max( j , j + 1). Then τ eβ0 , . . . , eβ j ⊆ eβζ j +1 ⊆ eβζ j +1 ⊆ eβ , since γ j ≤ β j ≤ βζ j and z γ j ⊆ β j+1 ⊆ βζ j . It follows that eβ ∈ X . Finally, let us show that eβ ∈ A(y ). Thus let α < π be such that yα ⊆ eβ . Then obviously yα ⊆ eη for some η < κ, so α ∈ e. There must be i < μ such that g(α) ∈ eβi . Then 

α ∈ g −1 ({g(α)}) ⊆ eβ , so α ∈ eβ .

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9 Weak saturation μ

In this section we are looking for cardinals θ such that N G κ,λ is not weakly θ -saturated. Lemma 9.1 (Solovay (see e.g. Lemma 8.8 in [17])) Let J be a normal ideal on κ such that E μκ ∈ J + . Then J is not weakly κ-saturated. Lemma 9.2 (i) (Matet [20]) Suppose that κ is a successor cardinal, and π is a cardinal such that Aκ,λ (κ, π ) holds. Then no κ-complete ideal on Pκ (λ) is weakly π -saturated. (ii) (Matet [22]) Suppose that κ is a limit cardinal, χ is a cardinal with κ ≤ χ ≤ λ, J is a χ -normal ideal on Pκ (λ), and X ⊆ Pκ (λ) is such that |X | > χ and {b ∈ Pκ (λ) : |X ∩ P(b)| ≤ |b ∩ χ |} ∈ J + . Then J is not weakly |X |-saturated. μ

Proposition 9.3 N G κ,λ is nowhere weakly λ-saturated. Proof Case λ = κ : By Propositions 2.3 and 2.6 and Lemma 9.1. Case when κ is a successor cardinal : By Lemma 9.2 (i). Case when κ is a limit cardinal and λ > κ : By Propositions 2.3 and 2.5 and Lemma 9.2 (ii). 

μ

If μ = ω, then by a result of [21], N G κ,λ is not weakly λℵ0 -saturated. This does μ not generalize since by Corollary 2.9 N G κ,λ is weakly λ+ -saturated in case ω < μ and λ < κ +ω . Lemma 9.4 (Usuba [34]) Suppose that cf(λ) < κ and J is a normal ideal on Pκ (λ) such that {a ∈ Pκ (λ) : cf(|a|) = cf(λ)} ∈ J + . Then J is not weakly λ+ -saturated. μ

Proposition 9.5 (i) Suppose that κ = ν + and cf(λ) ∈ κ\{cf(ν)}. Then N G κ,λ is nowhere weakly λ+ -saturated. μ (ii) Suppose that κ is a limit cardinal and cf(λ) ∈ κ\{μ}. Then N G κ,λ is nowhere weakly λ+ -saturated. 

Proof Use Lemma 9.4. μ

For other cases when cf(λ) < κ and N G κ,λ is nowhere weakly λ+ -saturated, see Propositions 5.2 and 5.3 (i) and Lemma 9.2 (i) if κ is a successor cardinal, and Proposition 5.3 (ii) and Lemma 9.2 (ii) otherwise. 

Note that by Corollary 6.8 and Lemma 9.2, if cf(λ) < κ, pp(λ) > λ+ and κ is a successor (respectively, limit) cardinal, then no κ-complete (respectively, normal) ideal J on Pκ (λ) is weakly λ++ -saturated. We conclude the section with a few more results asserting that under certain conμ ditions N G κ,λ is not weakly λ+ -saturated. Lemma 9.6 (Solovay [33]) N Sκ is nowhere weakly κ-saturated. Proposition 9.7 Suppose that κ is a successor cardinal and cf(λ) ∈ κ\{μ}. Then μ N G κ,λ is not weakly λ+ -saturated.

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Proof By Proposition 5.7 (i), I [λ+ ] contains a stationary subset S of E μλ . By an μ appeal to Corollary 7.5 we find h : Pκ (λ) −→ λ+ such that h(N G κ,λ ) ⊆ N Sλ+ | S. Lemma 9.6 tells us that there is Sα ∈ N Sλ+ ∩ P(S) for α < λ+ such that Sα ∩ Sβ = ∅ whenever α < β < λ+ . For α < λ+ , set Aα = h −1 (Sα ). Then {Aα : α < λ+ } ⊆ μ + (N G κ,λ ) . Moreover, Aα ∩ Aβ = ∅ whenever α < β < λ+ . 

Proposition 9.8 Suppose that cf(λ) = μ, and there is a successor cardinal κ such that (a) κ < κ < λ, (b) κ is not the successor of a cardinal of cofinality μ, and (c) μ cov(λ, κ , κ , κ) = λ. Then N G κ,λ is not weakly λ+ -saturated. μ

Proof By Proposition 2.15, there is g : Pκ (λ) −→ Pκ (λ) such that g(N G κ,λ ) = μ

μ

+

N G κ ,λ . Proposition 9.5 (i) tells us that there is Sα ∈ (N G κ ,λ ) for α < λ+ such that Sα ∩ Sβ = ∅ whenever α < β < λ+ . For α < λ+ , put Tα = g −1 (Sα ). Then μ + {Tα : α < λ+ } ⊆ (N G κ,λ ) . Furthermore, Tα ∩ Tβ = ∅ whenever α < β < λ+ .

Note that if cf(λ) < κ and γ + κ ≥ β for some γ < β, where ωβ = λ, then by Proposition 2.10 (i) there is a cardinal κ such that κ < κ < λ = cov(λ, κ , κ , κ) and κ is the successor of a regular cardinal. 10 Saturation μ

This final section is concerned with non-saturation of N G κ,λ . We start with the case where cf(λ) ≥ κ. Let us first recall the following general result. Lemma 10.1 (Foreman and Magidor [10]) Suppose that λ > κ, cf(λ) ≥ κ = ν + , where  cf(ν)  = μ, and J is a normal ideal on Pκ (λ) such that {a ∈ Pκ (λ) : cf(a ∩κ) = cf sup(a) = μ} ∈ J + . Then J is not λ+ -saturated. In particular, we have: Proposition 10.2 Suppose that λ > κ and cf(λ) ≥ κ = ν + , where cf(ν) = μ. Then μ N G κ,λ is nowhere λ+ -saturated. Let us next see what can be said when λ is a successor cardinal (and κ is not as in Proposition 10.2). Lemma 10.3 (Shelah (see e.g. [13])) Suppose that κ = ν + , where cf(ν) = μ. Then N Sκ | E μκ is nowhere κ + -saturated. μ

Proposition 10.4 Suppose that λ = θ + , where cf(θ ) = μ. Then N G κ,λ is not λ+ saturated. Proof By Proposition 5.7 (i), I [λ] contains a stationary subset S of E μλ . Appealing to Lemma 10.3 we find F ⊆ N Sλ+ ∩ P(S) with |F| = λ+ such that |W ∩ W | < λ for any two distinct members W, W of F. For W ∈ F, set TW = {a ∈ Pκ (λ) : sup(a) ∈ μ + W }. Then by Proposition 6.1 {TW : W ∈ F} ⊆ (N G κ,λ ) . It is simple to see that 

TW ∩ TW ∈ Iκ,λ whenever W, W ∈ F and W = W .

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λ Lemma 10.5 (Gitik and Shelah [13]) If λ is singular, then N Sλ+ | E cf(λ) is not λ++ saturated.

Proposition 10.6 Suppose that λ = θ + , where cf(θ ) = μ, and E μλ ∈ I [λ]. Then μ N G κ,λ is not λ+ -saturated. 

Proof Use Proposition 6.1 and Lemma 10.5.

Next we consider the case where λ is weakly inaccessible (and κ is not as in Proposition 10.2). Lemma 10.7 (Gitik and Shelah [13]) If κ is weakly inaccessible, then N Sκ | E μκ is not κ + -saturated. μ

Proposition 10.8 Suppose that λ is weakly inaccessible, and E μλ ∈ I [λ]. Then N G κ,λ is not λ+ -saturated. 

Proof The result follows from Proposition 6.1 and Lemma 10.7. μ N G κ,λ

If κ ≤ cf(λ) < λ and μ = ω, then by a result of [24] is not λ+ -saturated. We do not know whether this result remains valid in case μ > ω. What we can show is that if λ is a strong limit cardinal with κ ≤ cf(λ) < λ, and there is a [λ]<μ -normal μ ideal on Pκ (λ), then N G κ,λ is nowhere λ+ -saturated. The following is a straightforward generalization of a result of Foreman [8]. Lemma 10.9 Let J be a [λ]<μ -normal, (λ<μ )+ -saturated ideal on Pκ (λ). Then J is precipitous. Lemma 10.10 (Matet and Shelah [29]) Suppose that κ ≤ cf(λ) < λ, there exists a [λ]<μ -normal ideal on Pκ (λ), and τ cf(λ) ≤ λ<μ for every cardinal τ < λ. Then cf(λ) N Sκ,λ is nowhere precipitous. Proposition 10.11 Suppose that • • • •

κ ≤ cf(λ) < λ = cov(λ, κ, κ, μ+ ). There exists a [λ]<μ -normal ideal on Pκ (λ). τ cf(λ) ≤ λ<μ for every cardinal τ < λ.  [ρ]<μ  < λ for every cardinal ρ with There is a cardinal θ < λ such that cof N Sκ,ρ θ ∪ κ ≤ ρ < λ. μ

Then N G κ,λ is nowhere precipitous (and hence nowhere (λ<μ )+ -saturated). Proof By Proposition 4.12 (ii) and Lemmas 10.9 and 10.10.



The trouble with Proposition 10.11 is that it relies on pretty heavy cardinality assumptions. In this respect the following may look more attractive. Proposition 10.12 Suppose that there is a successor cardinal κ such that • κ < κ < λ. • κ ≤ cf(λ).

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• κ is not the successor of a cardinal of cofinality μ. • cov(λ, κ , κ , κ) = λ. μ

Then N G κ,λ is not λ+ -saturated. μ

Proof By Proposition 2.15, there is g : Pκ (λ) −→ Pκ (λ) such that N G κ ,λ = + μ By way of Proposition 10.2 we find Aα ∈ (N G κ ,λ ) for α < λ+ such μ that Aα ∩ Aβ ∈ N G κ ,λ for any two distinct members α, β of λ+ . For α < λ+ , set μ + Bα = g −1 (Aα ). Then clearly {Bα : α < λ+ } ⊆ (N G κ,λ ) . Moreover, Bα ∩ Bβ ∈ μ N G κ,λ whenever α < β < λ+ . 

μ g(N G κ,λ ).

μ

Corollary 10.13 Suppose that cov(λ, κ + , κ + , κ) = λ and cf(λ) > κ. Then N G κ,λ is not λ+ -saturated. Proof Use Proposition 10.4 if λ = κ + , and Proposition 10.12 (with κ = κ + ) otherwise. The remainder of the section is concerned with the case cf(λ) < κ. μ

Proposition 10.14 Suppose that κ > cf(λ) = μ. Then N G κ,λ is not Iκ,λ -λ++ - saturated. +

Proof By Proposition 5.7 (i) I [λ+ ] contains a stationary subset S of E μλ . By Lemma 10.3 we may find F ⊆ N Sλ++ ∩ P(S) with |F| = λ++ such that |W ∩W | < λ+ for any two distinct elements W, W of F. Now by Proposition 7.1 and Corollary 7.5, μ there is h : Pκ (λ) −→ λ+ such that Iλ+ ⊆ h(Iκ,λ ) and h(N G κ,λ ) ⊆ N Sλ+ | S. For μ

+

W ∈ F, put TW = h −1 (W ). Then clearly {TW : W ∈ F} ⊆ (N G κ,λ ) . Moreover given W, W ∈ F with W = W , TW ∩ TW ∈ Iκ,λ , since TW ∩ TW ⊆ h −1 (W ∩ W ) ∈ 

Iκ,λ . The following yields a stronger conclusion.

Proposition 10.15 Suppose that κ = ν + , where cf(ν) = μ, π > λ is a cardinal such μ ∗ that cf(π ) ≥ κ, and there is an Aκ,λ (κ, π )-sequence y such that A(y ) ∈ (N G κ,π ) . μ + Then N G κ,λ is nowhere Iκ,λ -π -saturated. Proof Let p : Pκ (π ) −→ Pκ (λ) be defined by p(x) = x ∩ λ. Now given B ∈ μ + μ + (N G κ,λ ) , set X = D y ∩ A(y ) ∩ p −1 (B). Then by Proposition 2.14 X ∈ (N G κ,π ) . μ

+

Hence by Proposition 10.2 we may find X α ∈ (N G κ,π ) ∩ P(X ) for α < π + such that X α ∩ X β ∈ Iκ,π whenever α < β < π + . For α < π + , put Bα = f y−1 (X α ). Since by   μ μ Lemma 8.2 and Proposition 8.4, N G κ,π = f y (N G κ,λ ) and Iκ,π ∩ P D y ∩ A(y ) ⊆ μ

+

f y (Iκ,λ ), {Bα : α < π + } ⊆ (N G κ,λ ) , and moreover Bα ∩ Bβ ∈ Iκ,λ whenever 

α < β < π +.

Proposition 10.15 can be combined with Propositions 8.6 and 8.7 to show that μ N G κ,λ is nowhere Iκ,λ -π + -saturated in a number of situations. Consider for instance the case where μ = ω1 , κ = ω4 and λ = ωω . Then by Proposition 5.2 Aκ,λ (μ, λ+ )

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holds, so by Propositions 8.6 and 10.15 N G κ,λ is nowhere λ++ -saturated. For another example, suppose that cf(λ) < μ, pp(λ) > λ+ and κ = ν + , where cf(ν) = μ. Then μ by Corollary 6.8 Aκ,λ (μ, λ++ ) holds, and hence by Propositions 8.6 and 10.15 N G κ,λ is nowhere λ+3 -saturated. Finally, let us deal with the case cf(λ) = μ. μ It has been shown [23] that if cf(λ) = μ = ω, then N G κ,λ is not Iκ,λ -λ++ -saturated. We do not know whether this holds in case μ = ω. The following is proved as Proposition 10.15. Proposition 10.16 Suppose that κ = ν + , where cf(ν) = μ, π > λ is a cardinal such μ + that cf(π ) ≥ κ, and there is an Aκ,λ (κ, π )-sequence y such that A(y ) ∈ (N G κ,π ) . μ + Then N G κ,λ is not Iκ,λ -π -saturated. Corollary 10.17 Suppose that κ = ν + , where cf(ν) = μ, and Bκ,λ (κ, π ) holds, where π is a cardinal greater than λ of cofinality greater than or equal to κ. Suppose μ further that I [κ] ∩ N Sκ+ ∩ P(E μκ ) = ∅. Then N G κ,λ is not Iκ,λ -π + -saturated. 

Proof By Propositions 8.9 and 10.16. Proposition 10.18 Suppose that there is a cardinal κ such that • κ < κ < λ. • κ = ν + , where cf(ν) = μ and I [κ ] ∩ N Sκ+ ∩ P(E μκ ) = ∅. • cov(λ, κ , κ , κ) = λ.

Suppose further that Bκ ,λ (κ , π ) holds, where π > λ is a cardinal with cf(π ) ≥ κ. μ Then N G κ,λ is not Iκ,λ -π + -saturated. Proof The proof mirrors that of Proposition 10.12, with Corollary 10.17 being used instead of Proposition 10.2. 

Corollary 10.19 Suppose that cf(λ) = μ > ω, cov(λ, κ + , κ + , κ) = λ and ADSλ μ holds. Then N G κ,λ is not Iκ,λ -λ++ -saturated. Proof Set κ = κ + . Then clearly, κ < κ < λ and moreover  by Proposition 5.7 (iii) E μκ ∈ I [κ ]. Since ADSλ holds, so does Bκ ,λ (cf(λ))+ , λ+ by Proposition 5.5. The desired conclusion is now immediate from Proposition 10.18. 

Acknowledgments

The author would like to thank the two referees for helpful suggestions.

References 1. Abraham, U., Magidor, M. : Cardinal arithmetic. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 2, pp. 1149–1227. Springer, Berlin (2010) 2. Cummings, J., Foreman, M., Magidor, M.: Squares, scales and stationary reflection. J. Math. Log. 1, 35–98 (2001) 3. Cummings, J., Foreman, M., Magidor, M.: Canonical structure in the universe of set theory I. Ann. Pure Appl. Log. 129, 211–243 (2004) 4. Dobrinen, N.: κ-stationary subsets of Pκ + λ, infinitary games, and distributive laws in Boolean algebras. J. Symb. Log. 73, 238–260 (2008)

123

P. Matet 5. Donder, H.D., Levinski, J.P.: Stationary sets and game filters, ca. (1987) (unpublished) 6. Donder, H.D., Matet, P.: Two cardinal versions of diamond. Isr. J. Math. 83, 1–43 (1993) 7. Eisworth, T.: Successors of singular cardinals. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 2, pp. 1229–1350. Springer, Berlin (2010) 8. Foreman, M.: Potent axioms. Trans. Am. Math. Soc. 294, 1–28 (1986) 9. Foreman, M., Magidor, M.: A very weak square principle. J. Symb. Log. 62, 175–196 (1997) 10. Foreman, M., Magidor, M.: Mutually stationary sequences of sets and the non-saturation of the nonstationary ideal on Pκ (λ). Acta Math. 186, 271–300 (2001) 11. Foreman, M., Todorcevic, S.: A new Löwenheim–Skolem theorem. Trans. Am. Math. Soc. 357, 1693–1715 (2005) 12. Gitik, M., Sharon, A.: On SCH and the approachability property. Proc. Am. Math. Soc. 136, 311–320 (2008) 13. Gitik, M., Shelah, S.: Less saturated ideals. Proc. Am. Math. Soc. 125, 1523–1530 (1997) 14. Holz, M., Steffens, K., Weitz, E.: Introductrion to Cardinal Arithmetic, Birkhäuser Advanced Texts : Basler Lehrbücher. Birkhäuser, Basel (1999) 15. Huuskonen, T., Hyttinen, T., Rautila, M.: On the κ-cub game on λ and I [λ]. Arch. Math. Log. 38, 549–557 (1999) 16. Huuskonen, T., Hyttinen, T., Rautila, M.: On potential isomorphism and non-structure. Arch. Math. Log. 43, 85–120 (2004) 17. Jech, T.J.: Set Theory, The Third Millenium Edition, Springer monographs in Mathematics. Springer, Berlin (2002) 18. Kueker, D.W.: Countable approximations and Löwenheim–Skolem theorems. Ann. Math. Log. 11, 57–103 (1977) 19. Matet, P.: Concerning stationary subsets of [λ]<κ . In: Stepr¯ans, J., Watson, S. (eds.) Set Theory and Its Applications, Lecture Notes in Mathematics, vol. 1401, pp. 119–127. Springer, Berlin (1989) 20. Matet, P.: Large cardinals and covering numbers. Fundam. Math. 205, 45–75 (2009) 21. Matet, P.: Game ideals. Ann. Pure Appl. Log. 158, 23–39 (2009) 22. Matet, P.: Weak saturation of ideals on Pκ (λ). Math. Log. Q. 57, 149–165 (2011) 23. Matet, P.: Non-saturation of the non-stationary ideal on Pκ (λ) with λ of countable cofinality. Math. Log. Q. 58, 38–45 (2012) 24. Matet, P.: Non-saturation of the non-stationary ideal on Pκ (λ) in case κ ≤ cf(λ) < λ. Arch. Math. Log. 51, 425–432 (2012) 25. Matet, P.: Normal restrictions of the noncofinal ideal on Pκ (λ). Fundam. Math. 221, 1–22 (2013) 26. Matet, P.: Two-cardinal diamond and games of uncountable length (preprint) 27. Matet, P., Péan, C., Shelah, S.: Cofinality of normal ideals on Pκ (λ) II. Isr. J. Math. 150, 253–283 (2005) 28. Matet, P., Péan, C., Shelah, S.: Cofinality of normal ideals on Pκ (λ) I (preprint) 29. Matet, P., Shelah, S.: The nonstationary ideal on Pκ (λ) for λ singular (preprint) 30. Mekler, A.H., Shelah, S.: Stationary logic and its friends. II. Notre Dame J. Form. Log. 27, 39–50 (1986) 31. Mitchell, W.J.: I [ω2 ] can be the nonstationary ideal on Cof(ω1 ). Trans. Am. Math. Soc. 361, 561–601 (2009) 32. Shelah, S.: Cardinal Arithmetic, Oxford Logic Guides, vol. 29. Oxford University Press, Oxford (1994) 33. Solovay, R.M.: Real-valued measurable cardinals. In: Scott, D.S. (ed.) Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematcis, vol. 13, part 1, American Mathematical Society, Providence, pp. 397–428 (1971) 34. Usuba, T.: Splitting stationary sets in Pκ λ for λ with small cofinality. Fundam. Math. 205, 265–287 (2009)

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