This paper explores how piece selection principles influence cardinal arithmetic, providing new results on distributivity and partition properties of certain ideals related to Shelah's work.
Contribution
It introduces novel connections between piece selection principles and cardinal arithmetic, addressing open questions by Abe and Usuba.
Findings
01
If λ ≥ 2^κ, then I_{κ, λ} is not (λ, 2)-distributive.
02
Under the same condition, I_{κ, λ}^+ does not satisfy a certain partition property.
03
The results extend understanding of the interplay between combinatorial principles and cardinal characteristics.
Abstract
We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if λ≥2κ, then (a) Iκ,λ is not (λ,2)-distributive, and (b) Iκ,λ+→(Iκ,λ+)ω2 does not hold.
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TopicsAdvanced Topology and Set Theory · Mathematical and Theoretical Analysis · Computability, Logic, AI Algorithms
Full text
PIECE SELECTION AND CARDINAL ARITHMETIC
Pierre MATET
Abstract
We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if λ≥2κ, then (a) Iκ,λ is not (λ,2)-distributive, and (b) Iκ,λ+→(Iκ,λ+)ω2 does not hold.
In [2] Abramson, Harrington, Kleinberg and Zwicker pointed out that many large cardinal properties can be reformulated as flipping properties, which are of the following type : One is given a family F of subsets of a set X. The property asserts that for some \sayflip h∈∏A∈F{A,X∖A}, the h(A)’s satisfy some intersection property (for instance the finite intersection property). \sayIntersection is taken in a wide sense so that e.g. diagonal intersections are allowed. Notice that with the family F is associated the family of all partitions of the form {A,X∖A}∖{∅} for A∈F. A flip of F can now be seen as a piece selection operation. Namely, for each partition {A,X∖A}∖{∅}, we choose one piece, either A or its complement.
For a typical example, let κ be a measurable cardinal, and U be a (κ-complete) measure on κ. For F take the collection of all partitions of κ into one or two pieces. For each such partition, select the piece in U. Then the pieces chosen have the property that the intersection of any less than κ many of them is cofinal in κ. Notice that since U is κ-complete, it does not matter whether our partitions have one, two or more pieces, as long as the number of pieces is less than κ. By thus increasing the number of pieces, we obtain a natural generalization of the original flipping properties. In this extended framework, regularity of an infinite cardinal κ can be expressed as the property that for any partition of κ into less than κ many pieces, one of the pieces must be cofinal in κ. The setting can be generalized by allowing J-partitions, and not just partitions. (Recall that for an ideal J on a set X, a J-partition of X is a subset Q of J+ such that
•
A∩B∈J for any two distinct members A,B of Q.
•
For any C∈J+, there is A∈Q with A∩C∈J+.)
This is a way to handle properties defined in terms of distributivity. For a further generalization we relax the requirement that a piece has to be selected in each partition. So for instance we’re given κ many partitions of κ, and we might be happy to pick one piece in κ many partitions. Piece selection principles of this type have been introduced in our joint paper [11] with Laura Fontanella. Their study is continued in the present paper.
Our starting point is Solovay’s celebrated result [33] on strongly compact cardinals and the Singular Cardinal Hypothesis. This theorem can be revisited in a number of ways. For instance it is shown in [22] that if cf(λ)<κ and there is a weakly λ++-saturated, (cf(λ))+-complete ideal on Pκ(λ), then pp(λ)=λ+. In another direction, Usuba [35] established that if κ is mildly λ-ineffable and cf(λ)≥κ, then λ<κ=λ, or equivalently (since κ is inaccessible), there is a cofinal subset of Pκ(λ) of size λ (i.e. u(κ,λ)=λ). Now it was noted [8] from the very beginning that mild ineffability can be reformulated as a piece selection principle. We consider various weakenings of this principle and attempt to compare their relative strengths. Some of these properties can be satisfied at a weakly, but not strongly, inaccessible cardinal, or even at a successor cardinal. So this part of the paper is a contribution to the age-old program of determining what’s left of weak or strong compactness when inaccessibility is removed. As the program developed, an impressive list of properties emerged, especially in connection with weak compactness. Some of these properties can be tentatively classified as weak (the tree property), of medium strength (our PS+) or strong (the weak compactness of the infinitary language Lκω). For others (e.g. our PS), the situation is not so clear and further work is needed. Our version of Solovay’s result reads as follows (see Proposition 3.4 and Observation 4.6).
THEOREM 1.1**.**
Suppose that cf(λ)<κ and PS+((cf(λ))+,κ,λ) holds. Then cov(λ,λ,(cf(λ))+,2)=λ+.
In the remainder of the paper, which is devoted to applications, we still deal with variants of mild ineffability, but this time inaccessibility of κ is implied. It is a central problem in the theory of Pκ(λ) to determine how the infinite Ramsey theorem generalizes in this framework. (Note that the theorem comes in several versions, from the weak \say{ω}→(Iω+)22 to the strong \sayIω+→(Iω+)mn for all finite n,m). The study of partition relations on Pκ(λ) is known to be tricky business. Carr [6] mentions that \sayrepeated efforts to obtain
•
{Pκ(λ)}→(Iκ,λ+)22 implies κ is mildly λ-ineffable,
•
κ is mildly λ-ineffable implies {Pκ(λ)}→(Iκ,λ+)23
\say
failed miserably. Johnson asked in [13] whether the (λ,2)-distributivity of Iκ,λ follows from the mild λ-ineffabilty of κ. This was answered in the negative by Abe [1] who showed that if λ<κ=2λ, then (a) Iκ,λ∣A is not (λ,2)-distributive for any stationary A, and (b) Iκ,λ+→(Iκ,λ+)22 does not hold. This led him to ask whether λ>κ implies that (a) Iκ,λ is not (λ,2)-distributive, and (b) Iκ,λ+→(Iκ,λ+)22 fails. This was answered, again in the negative, by Shioya [32]. It should be noted that his model is obtained by adding many Cohen subsets of κ. In fact it was shown in [17] that if κ is mildly λ<κ-ineffable and λ<κ<cov(Mκ,λ), then Iκ,λ+→(Iκ,λ+)ηn holds for any n<ω and any η<κ. Since cov(Mκ,λ)≤cov(Mκ,κ)≤dκ≤2κ, it made us think that maybe it could be proved that if λ≥2κ, then (a) Iκ,λ is not (λ,2)-distributive, and (b) Iκ,λ+→(Iκ,λ+)22 fails.
Part of the difficulty with the ordinary partition relation on Pκ(λ) stems from the fact that ⊂ is not a linear ordering. To avoid this kind of pitfalls we chose to work with weaker partition relations (note that by negating them, we will obtain stronger results). Given an ideal J on Pκ(λ) and a coloring of Pκ(λ)×Pκ(λ), we are looking for a color i and a set A in J+ that is not necessarily i-homogeneous (all pairs (a,b) from A×A with a⊂b have color i), but at least i-homogeneous mod J (meaning that for each a in A, the set of all b in A such that (a,b) does not have color i lies in J). We denote this particular partition relation by {Pκ(λ}J(J+)ρ2, where ρ denotes the number of available colors. Still weaker partition relations are obtained in a similar fashion by coloring κ×Pκ(λ) (given (a,b), we look at the color of (sup(a∩κ),b)) or κ+×Pκ(λ) (replace sup(a∩κ) with sup(a∩κ+)). To denote the corresponding partition property, we use Jκ (respectively, Jκ+). Proofs involve the usual ingredients (κ-normality, covering numbers, etc., and indeed some proofs are slight modifications of proofs of earlier results. Progress is achieved via a broader appeal to Shelah’s pcf theory.
Our efforts to prove the conjectures described above were only partially successful. By combining Observation 8.1 and Propositions 6.19, 8.4 and 9.6, one obtains the following.
THEOREM 1.2**.**
Suppose that 2κ≤λ, and let D∈NSκ,λ∗. Then setting J=Iκ,λ∣D, the following hold :
(i)
J+J(J+)ω2* does not hold.*
2. (ii)
J+J(J+)23* does not hold.*
3. (iii)
J* is not (λ,2)-distributive.*
There is a wide wide gap between this and what we can establish (see Proposition 5.32, Corollary 6.10, Observation 8.3 and Fact 9.5) under extra cardinal arithmetic assumptions such as Shelah’s Strong Hypothesis (SSH).
THEOREM 1.3**.**
Assuming SSH, the following hold :
(i)
If dκ≤λ and cf(λ)=κ, then for any D∈NSκ,λ∗, Iκ,λ∣D is not (κ,2)-distributive and (Iκ,λ∣D)+Iκ,λκ(Iκ,λ+,ω1)2 fails. If moreover κ is weakly Mahlo, then for any D∈NSκ,λ∗, (Iκ,λ∣D)+Iκ,λκ[Iκ,λ+]λ2 fails.
2. (ii)
If 2κ≤λ and cf(λ)=κ, then for any D∈NSκ,λ∗, Iκ,λ∣D is not (κ+,2)-distributive, and moreover (Iκ,λ∣D)+Iκ,λκ+[Iκ,λ+]λ2 fails.
On the positive side we have the following (see Corollary 9.2 and Fact 9.3).
THEOREM 1.4**.**
Suppose that SSH holds and either cf(λ)=κ, or cf(λ)<κ and λ+<dκ, or cf(λ)>κ and λ<dκ. Then the following hold :
(i)
Suppose that κ is weakly inaccessible. Then Iκ,λ+Iκ,λκ[Iκ,λ+]κ+2 holds.
2. (ii)
Suppose that κ is weakly compact. Then Iκ,λ is (κ,2)-distributive, and moreover Iκ,λ+Iκ,λκ(Iκ,λ+)η2 holds whenever 0<η<κ.
Concerning Iκ,λ+→(Iκ,λ+)22, it remains open whether it fails whenever 2κ≤λ. What we do know is that it fails if λ is large enough. In fact as shown in [18], {Pκ(λ)}Iκ,λ[Iκ,λ+]λ2 fails if λ is large enough.
The article is organized as follows. Section 2 is devoted to piece selection principles on Pκ(λ), with emphasis on PS+(τ,κ,λ). It is shown that if cf(λ)<κ and PS+((cf(λ))+,κ,λ) holds, then there is no remarkably good scale on λ. Section 3 is concerned with piece selection principles on κ. It is observed that the tree property TP(κ) is one of them. We use scales to establish that if cf(λ)<κ, then PS+((cf(λ))+,κ,λ) implies PS+((cf(λ))+,λ+). Section 4 is devoted to Shelah’s covering numbers. It is shown that if λ is singular and PS+((cf(λ))+,λ+) holds, then cov(λ,λ,(cf(λ))+,2)=λ+. In Section 5 we give cardinal arithmetic conditions under which for any club subset C of Pκ(λ), the partition property (Iκ,λ∣C)+Iκ,λκ(Iκ,λ+,ρ)2 fails. This is continued in Section 6 where we deal with the stronger partition relations (Iκ,λ∣C)+Iκ,λκ(Iκ,λ+)22 and (Iκ,λ∣C)+Iκ,λ(Iκ,λ+)ω2. Mild ineffability is the subject of Section 7. We prove that if κ is mildly λ-ineffable and cf(λ)=κ, then cov(λ,κ+,κ+,κ)=λ. Section 8 contains results on the non-distributivity of Iκ,λ∣C for a club subset C of Pκ(λ). Finally in Section 9, we deal with the remaining case, that is the case when cf(λ)=κ, and explain why this case must be handled separately.
2 Piece selection
Throughout the paper κ will denote a regular uncountable cardinal, and λ a cardinal greater than or equal to κ. We start with some definitions.
DEFINITION 2.1**.**
For a set A and a cardinal τ, we set Pτ(A)={a⊆A:∣a∣<τ} and [A]τ={x⊆A:∣x∣=τ}.
DEFINITION 2.2**.**
By a partition of a set X we mean a subset Q of P(X)∖{∅} such that:
•
A∩B=∅ for any two distinct members A,B of Q.
•
⋃Q=X.
DEFINITION 2.3**.**
An ideal on a set X is a nonempty collection J of subsets of X such that :
•
A∪B∈J whenever A,B∈J.
•
P(A)⊆J for all A∈J.
•
X∈/J.
Given an ideal J on X, we denote by J+ the set {A⊆X:A∈/J}, while J∗ denotes the set {A⊆X:X∖A∈J}. For any A∈J+, we let J∣A={B⊆X:B∩A∈J}.
We say that J is κ-complete if for any collection Z of less than κ many sets in J, one has ⋃Z∈J.
An ideal K on XextendsJ if J⊆K.
We let Iκ=⋃α<κP(α) and
Iκ,λ=⋃a∈Pκ(λ)P({b∈Pκ(λ):a∖b=∅}).
An ideal J on κ (respectively, Pκ(λ)) is fine if it extends Iκ (respectively, Iκ,λ).
We let NSκ (respectively, NSκ,λ) denote the nonstationary ideal on κ (respectively, Pκ(λ)).
DEFINITION 2.4**.**
Let τ be an infinite cardinal less than or equal to κ. The piece selection principle PS+(τ,κ,λ) means that given a partition Qa of Pκ(λ) with ∣Qa∣<τ for each a∈Pκ(λ), there is B∈Iκ,λ+ and h∈∏a∈Pκ(λ)Qa such that for any a,b∈B, the set {c∈h(a)∩h(b):a∪b⊆c} is nonempty.
PS∗(τ,κ,λ)) (respectively, PS(τ,κ,λ)) means that given a partition Qa of Pκ(λ) with ∣Qa∣<τ for each a∈Pκ(λ), we may find B∈Iκ,λ+ and h∈∏a∈Pκ(λ)Qa such that for any a,b∈B, there is t in B (respectively, in Pκ(λ)) such that a∪b⊆t and the two sets {c∈h(a)∩h(t):t⊆c} and {d∈h(b)∩h(t):t⊆d} are nonempty.
OBSERVATION 2.5**.**
PS+(τ,κ,λ)⇒PS∗(τ,κ,λ)⇒PS(τ,κ,λ).
OBSERVATION 2.6**.**
Suppose that PS∗(κ,κ,λ) holds. Then κ is weakly inaccessible.
Proof. Suppose otherwise, and let κ=ν+. For ν≤γ<κ, select a bijection jγ:γ→ν. Put A={a∈Pκ(λ):ν⊆a}. For a∈A and i<ν, let Qai denote the collection of all c∈Pκ(λ) such that a⊆c and j(sup(c∩κ))+1(sup(a∩κ))=i. We may find B∈Iκ,λ+∩P(A) and i<ν such that for any a,b∈B, there is t in B such that a∪b⊆t and the two sets {c∈Qai∩Qti:t⊆c} and {d∈Qbi∩Qti:t⊆d} are nonempty. Now pick a,b∈B with sup(a∩κ)<sup(b∩κ). There must be t in B with a∪b⊆t, c∈Qai∩Qti with t⊆c and d∈Qbi∩Qti with t⊆d. But then
It follows that sup(a∩κ)=sup(t∩κ)=sup(b∩κ). Contradiction.
□
OBSERVATION 2.7**.**
Suppose that PS+(τ,κ,λ) holds. Let A∈Iκ,λ+, and for each a∈A, let Qa be a partition of the set {c∈A:a⊆c} with ∣Qa∣<τ. Then there is B∈Iκ,λ+∩P(A) and h∈∏a∈BQa such that for any a,b∈B, h(a)∩h(b)=∅.
Proof. Define ψ:Pκ(λ)→A so that x⊆ψ(x) for all x∈Pκ(λ). For x∈Pκ(λ), let Tx denote the set of all z∈Pκ(λ) such that x⊆z but ψ(x)∖z=∅, and set
Zx={{z∈Pκ(λ):ψ(x)⊆z and ψ(z)∈W}:W∈Qψ(x)}.
We may find H∈Iκ,λ+ and k∈∏x∈Pκ(λ)(Zx∪{Tx}) such that k(x)∩k(y)=∅ for any x,y∈H.
Claim. Let x∈H. Then k(x)∈Zx.
Proof of the claim. Suppose otherwise. Pick y∈H with ψ(x)⊆y, and z∈k(x)∩k(y). Then ψ(x)⊆y⊆z. This contradiction completes the proof of the claim.
Now put B=ψ‘‘H, and define f:B→H such that ψ(f(a))=a for all a∈B. Notice that B∈Iκ,λ+∩P(A). Let h∈∏a∈BQa be such that for any a∈B,
k(f(a))={z∈Pκ(λ):ψ(f(a))⊆z and ψ(z)∈h(a)}.
Given a,b∈B, we may find z in k(f(a))∩k(f(b)). Then a∪b=ψ(f(a))∪ψ(f(b))⊆z⊆ψ(z), and moreover ψ(z)∈h(a)∩h(b).
□
OBSERVATION 2.8**.**
Suppose that PS+(τ,κ,λ) holds. Then for any cardinal χ with κ≤χ<λ, PS+(τ,κ,χ) holds.
Proof. Let χ be a cardinal with κ≤χ<λ, and for each y∈Pκ(χ), let Qy be a partition of Pκ(χ) with ∣Qy∣<τ. For a∈Pκ(λ), put
Wa={{c∈Pκ(λ):c∩χ∈Z}:Z∈Qa∩χ}.
Note that {c∩χ:c∈S}∈Qa∩χ for all S∈Wa. We may find B∈Iκ,λ+ and h∈∏a∈Pκ(λ)Wa such that for any a,b∈B,
{c∈h(a)∩h(b):a∪b⊆c}=∅.
Set Y={b∩χ:b∈B}. Notice that Y∈Iκ,χ+. Select ψ:Y→B so that for any y∈Y, y=ψ(y)∩χ, and define k∈∏y∈YQy by k(y)={c∩χ:c∈h(ψ(y))}. Now given x,y∈Y, pick c∈h(ψ(x))∩h(ψ(y)) with ψ(x)∪ψ(y)⊆c. Then clearly, x∪y⊆c∩χ, and moreover c∩χ∈k(x)∩k(y).
□
Let us next recall some material concerning scales in pcf theory.
DEFINITION 2.9**.**
Let A be an infinite set of regular cardinals such that ∣A∣<minA, and I be an ideal on A such that {A∩a:a∈A}⊆I.
We let ∏A=∏a∈Aa. For f,g∈∏A, we let f<Ig if {a∈A:f(a)≥g(a)}∈I.
Let π be a regular cardinal greater than supA. An increasing, cofinal sequence f=⟨fα:α<π⟩ in (∏A,<I) is said to be a scale of lengthπ. If there is such a sequence, we set tcf(∏A/I)=π.
FACT 2.10**.**
([30, Theorem 1.5 p. 50])* Suppose that λ is a singular cardinal. Then there is a set A of regular cardinals such that o.t.(A)=cf(λ)<minA,supA=λ and tcf(∏A/I)=λ+, where I is the noncofinal ideal on A.*
DEFINITION 2.11**.**
Let f=⟨fα:α<π⟩ be an increasing, cofinal sequence in (∏A,<I). An infinite limit ordinal δ<π is a good (respectively, remarkably good) point for f if there is a cofinal (respectively, closed unbounded) subset X⊆δ, and Zξ∈I for ξ∈X such that fβ(a)<fξ(a) whenever β<ξ are in X and a∈A∖(Zβ∪Zξ).
δ is a better point for f if we may find a closed unbounded subset X of δ, and Zξ∈I for ξ∈X such that fβ(a)<fξ(a) whenever β<ξ are in X and a∈A∖Zξ.
δ is a very good point for f if there is a closed unbounded subset X of δ, and Z∈I such that fβ(a)<fξ(a) whenever β<ξ are in X and a∈A∖Z.
The scale f=⟨fα:α<π⟩ is good (respectively, remarkably good, better, very good) if there is a closed unbounded subset C of π with the property that every infinite limit ordinal δ in C such that cf(δ)<supA and I is not cf(δ)-complete is a good (respectively, remarkably good, better, very good) point for f.
FACT 2.12**.**
([7], [20])* Let δ<π be an infinite limit ordinal such that I is cf(δ)-complete (respectively, (cf(δ))+-complete). Then δ is a better (respectively, very good) point for f.*
We will show that if cf(λ)<κ and PS+((cf(λ))+,κ,λ+) holds, then there is no remarkably good scale on λ.
DEFINITION 2.13**.**
Given two infinite cardinals τ and χ such that τ≤χ=cf(χ), we let Eτχ (respectively, E<τχ) denotes the set of all infinite limit ordinals α<χ such that cf(α)=τ (respectively, cf(α)<τ).
OBSERVATION 2.14**.**
Suppose that tcf(∏A/I)=π, where
•
A* is an infinite set of regular cardinals such that ∣A∣<minA,*
•
I* is an ideal on A such that {A∩a:a∈A}⊆I,*
and f=⟨fα:α<π⟩ is an increasing, cofinal sequence in (∏A,<I). Let χ be an infinite cardinal with χ≤(supA)+. Then the following are equivalent :
(i)
There is a closed unbounded subset C of π such that for any regular infinite cardinal θ<χ, and any δ∈C∩Eθπ, δ is a remarkably good point for f.
2. (ii)
There is a closed unbounded subset D of π such that for any e∈Pχ(D), there is g:e→I such that fα(a)<fβ(a) whenever α<β are in e and a∈A∖(g(α)∪g(β)).
Proof. (i) → (ii) : By Proposition 8.11 of [21] (we can take D=C).
(ii) → (i) : Easy (take C= the set of limit points of D).
□
PROPOSITION 2.15**.**
Suppose that tcf(∏A/I)=λ+, where
•
A* is a set of regular cardinals such that ∣A∣<min{κ,minA} and supA=λ.*
•
I* is an ideal on A such that {A∩a:a∈A}⊆I.*
•
PS+(∣A∣+,κ,λ+)* holds.*
Let f=⟨fα:α<λ+⟩ be an increasing, cofinal sequence in (∏A,<I), and let S denote the set of all infinite limit points δ<λ+ such that
•
cf(δ)<κ.
•
δ* is not a remarkably good point for f.*
Then S is stationary in λ+.
Proof. Suppose otherwise, and select a closed unbounded subset C of λ+ such that any infinite limit ordinal δ in C of cofinality less than κ is a remarkably good point for f. By Fact 2.14, for any e∈Pκ(C), there is ge:e→I such that fα(a)<fβ(a) whenever α<β are in e and a∈A∖(ge(α)∪ge(β)). Pick a bijection k:λ+→C and for each nonempty b∈Pκ(λ+), tb∈∏α∈k‘‘b(A∖gk‘‘b(α)). Put X={x∈Pκ(λ+):supx∈x}. For x∈X and a∈A, set
Qxa={b∈X:x⊆b and tb(k(supx))=a}.
By Observation 2.7 we may find B∈Iκ,λ+∩P(X) and h:B→A such that for any x,y∈B, Qxh(x)∩Qyh(y)=∅. There must be H∈Iκ,λ++∩P(B) and a∈A such that h takes the constant value a on H. Put D={k(supx):x∈H}.
Claim. Let α<β in D. Then fα(a)<fβ(a).
Proof of the claim. Pick x,y∈H with k(supx)=α and k(supy)=β, and b∈Qxa∩Qya. Then tb(α)=a=tb(β). Thus a∈A∖(gk‘‘b(α)∪gk‘‘b(β)), and therefore fα(a)<fβ(a), which completes the proof of the claim.
By the claim, the function u:D→A defined by u(α)=fα(a) is one-to-one. Contradiction.
□
Let us observe that by replacing the hypothesis that PS+(∣A∣+,κ,λ+) holds with the stronger hypothesis that κ is mildly λ+-ineffable, we can actually prove [20] that for cofinally many regular uncountable cardinals σ<κ, the set of all points δ∈Eσλ+ that are not remarkably good for f is stationary.
The conclusion of Proposition 2.15 entails the failure of a square principle of the following type.
DEFINITION 2.16**.**
Let θ and χ be two infinite cardinals such that θ≤χ=cf(χ), and S⊆E<χχ. Then WWSχθ(S) asserts the existence of Cγ for γ<χ such that
•
∣Cγ∣<χ ;
•
Cγ⊆Pθ(γ) ;
•
if α∈S, then there is a closed unbounded subset C of α of order type cf(α) such that C∩γ∈⋃D∈CγP(D) for every γ∈C.
Further let T be a collection of regular cardinals such that for any σ∈T,
•
σ<supA* ;*
•
WWSπsupA(Eσπ)* holds.*
Then there is an increasing, cofinal sequence f=⟨fα:α<π⟩ in (∏A,<I) such that for any σ∈T, and any ζ∈Eσπ, ζ is a better point for f.
PROPOSITION 2.18**.**
Suppose that cf(λ)<κ and PS+((cf(λ))+,κ,λ+) holds. Then WWSλ+λ(Eσλ+) fails for some regular cardinal σ with cf(λ)<σ<κ.
Proof. By Proposition 2.15 and Facts 2.10, 2.12 and 2.17.
□
We next consider the two-cardinal version of the tree property.
DEFINITION 2.19**.**
TP(κ,λ) asserts the following. Let sa⊆a for a∈Pκ(λ) be such that for some C∈NSκ,λ∗, we have ∣{sa∩c:c⊆a}∣<κ for all c∈C. Then there is S⊆λ with the property that for every b∈Pκ(λ), there is a∈Pκ(λ) such that b⊆a and S∩b=sa∩b.
FACT 2.20**.**
(([37])* It is consistent relative to a supercompact cardinal that TP(ω2,χ) holds for every cardinal χ≥ω2.*
OBSERVATION 2.21**.**
Let sa⊆a for a∈Pκ(λ). Suppose that there is C∈Iκ,λ+ such that ∣{sa∩c:c⊆a}∣<κ for all c∈C. Then ∣{sa∩d:d⊆a}∣<κ for all d∈Pκ(λ).
Proof. Suppose otherwise. Then we may find d∈Pκ(λ) and ai∈Pκ(λ) for i<κ so that
•
d⊆ai for all i<κ.
•
sai∩d=saj∩d whenever i<j<κ.
Pick c∈C with d⊆c, and let i<j<κ. Then sai∩c=saj∩c, since otherwise we would have sai∩d=saj∩d. Contradiction.
□
FACT 2.22**.**
PS(κ,κ,λ)* implies TP(κ,λ).*
Proof. By Proposition 5.4 of [11] and Observation 2.20.
□
We will now see that if in the definition of PS+ we insist on selecting a piece from every partition, then what we obtain is an apparently much stronger principle.
DEFINITION 2.23**.**
For a fine ideal J on Pκ(λ), the ideal extension principleIE(τ,κ,λ,J) means that given a partition Qa of Pκ(λ) with ∣Qa∣<τ for each a∈Pκ(λ), there is h∈∏a∈Pκ(λ)Qa and an ideal K on Pκ(λ) extending J such that ran(h)⊆K∗.
OBSERVATION 2.24**.**
IE(ω,κ,λ,J)* holds.*
Proof. Let Qa be a finite partition of Pκ(λ) for a∈Pκ(λ). Select a prime ideal K extending J. For each a∈Pκ(λ), there must be Wa∈Qa with Wa∈K∗. Now define h∈∏a∈Pκ(λ)Qa by h(a)=Wa.
□
FACT 2.25**.**
([25]* There is a partition of Pκ(λ) into λ<κ sets in Iκ,λ+.*
OBSERVATION 2.26**.**
The following are equivalent :
(i)
IE(τ,κ,λ,Iκ,λ)* holds.*
2. (ii)
*Given a partition Qa of Pκ(λ) with ∣Qa∣<τ for each a∈Pκ(λ), there is h∈∏a∈Pκ(λ)Qa such that for any a,b∈Pκ(λ), there is c∈h(a)∩h(b) with a∪b⊆c.
*
Proof. (i) → (ii) : Trivial.
(ii) → (i) : Assume that (ii) holds, and let Qa be a partition of Pκ(λ) with ∣Qa∣<τ for each a∈Pκ(λ). By Fact 2.25, we may find a partition T of Pκ(λ) into λ<κ sets in Iκ,λ+. Let ⟨Tw:w∈Pκ(λ)⟩ be a one-to-one enumeration of T. Pick a bijection F:Pκ(λ)→Pω(Pκ(λ))∖{∅}. For w∈Pκ(λ) and x∈Tw, put Wx={⋂a∈F(w)k(a):k∈∏a∈F(w)Qa}. We may find g∈∏x∈Pκ(λ)Wx such that for any x,y∈Pκ(λ), there is z∈g(x)∩g(y) with x∪y⊆z.
For w∈Pκ(λ) and x∈Tw, let kx∈∏a∈F(w)Qa be such that g(x)=⋂a∈F(w)kx(a).
Claim 1. Let v,w∈Pκ(λ), x∈Tv, y∈Tw and a∈F(v)∩F(w). Then kx(a)=ky(a).
Proof of Claim 1. Suppose otherwise. Then kx(a)∩ky(a)=∅. Since g(x)⊆kx(a) and g(y)⊆ky(a), it follows that g(x)∩g(y)=∅. This contradiction completes the proof of the claim.
Put h=⋃x∈Pκ(λ)kx. Using Claim 1, it is easy to see that h∈∏Qa.
Claim 2. Let e∈Pω(Pκ(λ))∖{∅}. Then ⋂a∈eh(a)∈Iκ,λ+.
Proof of Claim 2. Let e=F(w). Now given s∈Pκ(λ), pick x∈Tw with s⊆x. There must be z∈⋂a∈F(w)kx(a) with x⊆z. Then clearly, s⊆z, and moreover z∈⋂a∈eh(a). This completes the proof of the claim and that of the observation.
□
3 Piece selection at κ
In this section we concentrate on the case λ=κ.
DEFINITION 3.1**.**
Given an infinite cardinal τ, we let PS+(τ,κ) assert the following: For β∈κ, let Qβ be a partition of κ∖β with ∣Qβ∣<τ. Then there is a cofinal subset B of κ and h∈∏β∈BQβ such that for any α,β∈B, we have h(α)∩h(β)=∅.
PS∗(τ,κ) (respectively PS(τ,κ)) asserts the following: For β∈κ, let Qβ be a partition of κ∖β with ∣Qβ∣<τ. Then we may find a cofinal subset B of κ and h∈∏β∈κQβ such that for any α,β∈B, there is ζ in B (respectively, in κ) such that max{α,β}≤ζ and we have h(α)∩h(ζ)=∅ and h(β)∩h(ζ)=∅.
OBSERVATION 3.2**.**
The following are equivalent :
(i)
PS+(τ,κ).
2. (ii)
PS+(τ,κ,κ).
Proof. (i) → (ii) : Suppose that (i) holds. For b∈Pκ(κ), let Qb be a partition of the set {c∈Pκ(κ):b⊆c} into less than τ many pieces. For b∈Pκ(κ), put ρb=∣Qb∣ and let ⟨Qbi:i<ρb⟩ be a one-to-one enumeration of Qb. Now for β<κ and i<ρβ, set Wβi=Qβi∩κ. We may find a cofinal subset B of κ and h∈∏β∈κρβ such that for any α,β∈B, we have Wαh(α)∩Wβh(β)=∅. Then clearly, B∈Iκ,κ+, and moreover Qαh(α)∩Qβh(β)=∅ for all α,β∈B.
□
(ii) → (i) : By Observation 2.7.
□
OBSERVATION 3.3**.**
(i)
PS∗(τ,κ)* implies PS∗(τ,κ,κ).*
2. (ii)
PS(τ,κ)* implies PS(τ,κ,κ).*
Proof. Argue as for Observation 3.2.
□
COROLLARY 3.4**.**
PS(κ,κ)* implies TP(κ,κ).*
Proof. Use Fact 2.22.
□
PROPOSITION 3.5**.**
(i)
Suppose that λ is regular and PS+(τ,κ,λ) holds. Then PS+(τ,λ) holds.
2. (ii)
Suppose that cf(λ)<κ and PS+((cf(λ))+,κ,λ) holds. Then PS+((cf(λ))+,λ+) holds.
Proof. (i) : For α<λ, let Qα be a partition of λ∖α into less than τ many pieces. For α∈λ, put ρα=∣Qα∣ and let ⟨Qαi:i<ρα⟩ be a one-to-one enumeration of Qα. Now for a∈Pκ(λ) and i<ρsupa, set
Wai={c∈Pκ(λ):a⊆c and supc∈Qsupai}.
We may find B∈Iκ,λ+ and h∈∏a∈Bρsupa such that for any a,b∈B, we have Wah(a)∩Wbh(b)=∅. Put A={supa:a∈B}, and pick ψ:A→B so that sup(ψ(α))=α for all α∈A. Notice that A∈Iλ+. Given α,β∈A, pick c∈Wψ(α)h(ψ(α))∩Wψ(β)h(ψ(β)). Then clearly, supc∈Qαh(ψ(α))∩Qβh(ψ(β)).
(ii) : Put cf(λ)=σ. For ξ<λ+, let Wξ be a partition of λ+∖ξ into at most σ many pieces. For ξ∈λ+, put ρξ=∣Wξ∣ and let ⟨Wξδ:δ<ρξ⟩ be a one-to-one enumeration of Wξ. By Fact 2.10, we may find an increasing sequence ⟨λi:i<σ⟩ of regular cardinals greater than κ with supremum λ, and an increasing cofinal sequence ⟨fα:α<λ+⟩ in (∏i<σλi,<∗), where f<∗g just in case ∣{i<σ:f(i)≥g(i)}∣<σ. For a∈Pκ(λ), define χa∈∏i<σλi by χa(i)=sup(a∩λi). Define s:Pκ(λ)→λ+ by : s(a)= the least α such that χa<∗fα. For a∈Pκ(λ) and δ<σ, let Qaδ denote the collection of all c∈Pκ(λ) such that a⊆c and s(c)∈Ws(a)δ. We may find B∈Iκ,λ+ and h∈∏a∈Bρs(a) such that for any a,b∈B, we have Qah(a)∩Qbh(b)=∅.
We inductively define an∈B for n<λ+ so that s(am)<s(an) whenever m<n<λ+. Suppose that am has been defined for each m<n. Putting γ=sup{s(am):m<n}, we select an∈B so that ran(fγ+1)⊆an. Now let m<n<λ+ be given. There must be some c in Qamh(am)∩Qanh(an). Then clearly, s(c)≥max{s(am),s(an)}, and moreover s(c)∈Ws(am)h(am)∩Ws(an)h(an).
□
DEFINITION 3.6**.**
Given an infinite cardinal χ, the Almost Disjoint Set principleADSχ asserts the existence of a cofinal subset yα of χ of order-type cf(χ) for each α<χ+ such that for each nonzero β<χ+, there is k∈∏α<βyα with the property that (yδ∖(k(δ))∩(yα∖(k(α))=∅ whenever δ<α<β.
It is known [21] that if there is a remarkably good scale on χ, then ADSχ holds. Thus the following is closely related to Proposition 2.15.
PROPOSITION 3.7**.**
Let χ be a singular cardinal such that PS+((cf(χ))+,χ) holds. Then ADSχ fails.
Proof. Let yα be a cofinal subset of χ of order-type cf(χ) for each α<χ+. Suppose that for each nonzero β<χ+, there is kβ∈∏α<βyα with the property that (yδ∖(kβ(δ))∩(yα∖(kβ(α))=∅ whenever δ<α<β. For α<χ+ and ξ∈yα, let Qαξ denote the set of all γ such that α≤γ<χ+ and kγ+1(α)=ξ. We may find B∈[χ+]χ+ and h∈∏α∈Byα such that Qδh(δ)∩Qαh(α)=∅ whenever δ,α∈B. Now given δ<α<χ+, pick γ∈Qδh(δ)∩Qαh(α). Then clearly, kγ+1(δ)=h(δ) and kγ+1(α)=h(α), and consequently (yδ∖(h(δ))∩(yα∖(h(α))=∅. Contradiction.
□
Let us now turn to the tree property.
DEFINITION 3.8**.**
The tree propertyTP(κ) asserts that any tree of height κ each of whose levels has size less than κ has a κ-branch.
FACT 3.9**.**
(i)
([37])* TP(κ) and TP(κ,κ) are equivalent.*
2. (ii)
([34])* Let τ be an infinite cardinal such that τ<τ=τ. Then TP(τ+) fails.*
TP(κ) can be recast as a piece selection principle.
OBSERVATION 3.10**.**
The following are equivalent:
(i)
TP(κ).
2. (ii)
Let ⟨Qα:α<κ⟩ be a sequence of partitions of κ into less than κ many pieces with the property that Qβ⊆⋃W∈QαP(W) whenever α<β<κ. Then there is h∈∏α<κQα such that ∣h(α)∩h(β)∣≥2 whenever α<β<κ.
Proof. (i) → (ii) : Suppose that (i) holds, and let ⟨Qα:α<κ⟩ be as in (ii). Consider the tree (T,<T), where
•
T=⋃γ<κLγ, where Lγ consists of all g∈∏α<γ+1{A∈Qα:∣A∣≥2} such that g(β)⊆g(α) whenever α<β<γ.
•
f<Tg just in case f⊂g.
(ii) → (i) : Suppose that (ii) holds, and let T=(κ,<T) be a tree of height κ with each level Lα of size less than κ. Consider the sequence ⟨Qα:α<κ⟩ of partitions of κ defined by : Qα={Qαξ:ξ∈Lα}, where
Qαξ={ζ∈κ:ζ<Tξ}∪{ξ}∪{η∈κ:ξ<Tη}.
□
This can be used to reformulate TP(κ) in terms of partitions relations.
OBSERVATION 3.11**.**
The following are equivalent:
(i)
TP(κ).
2. (ii)
Suppose that F:κ×κ→κ has the following property : if β<γ<δ<κ are such that F(β,γ)=F(β,δ), then F(α,γ)=F(α,δ) for all α<β. Then there is A∈[κ]κ such that one of the following holds :
•
F(β,γ)=F(β,δ)* whenever β<γ<δ are in A.*
•
F(β,γ)=F(β,δ)* whenever β<γ<δ are in A.*
Proof. (i) → (ii) : Assume that (i) holds, and let F be as in (ii). For α<κ, consider the equivalence relation ∼α defined on κ by : β∼αγ if and only if either γ=β≤α, or β,γ>α and F(ξ,β)=F(ξ,γ) for all ξ≤α. Let Qα be the set of all equivalence classes with respect to ∼α.
Case 1 : There is η<κ such that ∣Qη∣=κ. Pick A∈[κ∖(η+1))]κ so that ∣A∩H∣≤1 for all H∈Qη. Now if β<γ<δ are in A, we must have F(β,γ)=F(β,δ), since otherwise we would have F(η,γ)=F(η,δ).
Case 2 : ∣Qα∣<κ for all α<κ. Then by Observation 3.10, we may find h∈∏α<κQα such that ∣h(α)∩h(β)∣≥2 whenever α<β<κ. It is simple to see that {h(α):α<κ}⊆[κ]κ. Furthermore, h(β)⊆h(α) whenever α<β<κ. Now inductively define an increasing sequence ⟨αi:i<κ⟩ of elements of κ so that for any j<κ, αj∈h((sup{αi:i<j})+1). Then clearly, F(αi,αj)=F(αi,αk) whenever i<j<k<κ.
(ii) → (i) : Assume that (ii) holds, and let ⟨Qα:α<κ⟩ be as in (ii) of Observation 3.10. For α<κ, let ⟨Qαi:i<∣Qα∣⟩ be a one-to-one enumeration of Qα. Define F:κ×κ→κ by : F(α,β)=i just in case β∈Qαi. There must be A∈[κ]κ and f∈∏α∈A∣Qα∣ such that F(β,γ)=f(β) whenever β<γ are in A. Then A∖(β+1)⊆Qβf(β) for all β∈A. It easily follows that the conclusion of (ii) of Observation 3.10 holds.
□
QUESTION. Is it consistent that TP(κ) holds, but PS(κ,κ) fails ?
We return to ideal extension, but this time for ideals on κ.
DEFINITION 3.12**.**
We let Lκω denote the infinitary language which allows conjunctions and disjunctions of less than κ many formulas, and universal and existential quantification over finitely many variables.
Lκω is weakly compact if any set of κ sentences from Lκω without a model has a subset of smaller size without a model.
DEFINITION 3.13**.**
For a fine ideal J on κ, IE(τ,κ,J) means that given a partition Qα of κ∖α with ∣Qα∣<τ for each α∈κ, there is h∈∏α∈κQα and an ideal K on κ extending J such that ran(h)⊆K∗.
OBSERVATION 3.14**.**
Suppose that Lκω is weakly compact. Then IE(κ,κ,Iκ) holds.
Proof. For α<κ, let Qα be a partition of κ∖α with ∣Qα∣<κ. Consider the Lκω language with one unary predicate S and constant symbols cA for A∈⋃α<κQα. Let Σ consist of the following sentences :
•
⋁A∈QαS(cA) for each α<κ.
•
¬(S(cA0)∧S(cA1)∧⋯∧S(cAn)) whenever 0<n<ω, A0,A1,⋯,An∈⋃α<κQα and A0∩A1∩⋯∩An=∅.
Notice that for 0<β<κ, ⋂α<βkβ(α)=∅, where kβ:β→κ is defined by kβ(α)= the unique A∈Qα such that β∈A. It easily follows that any subset of Σ of size less than κ is satisfiable. Hence so is Σ itself, and there must be h∈∏α<κQα with the property that for each e∈Pω(κ)∖{∅}, ⋂α∈eh(α) is nonempty. In fact, ⋂α∈eh(α)∈Iκ+. Suppose otherwise, and let δ<κ such that ⋂α∈eh(α)⊆δ. Then ⋂α∈dh(α)=∅, where d=e∪{δ}. Contradiction.
□
Boos [4] showed that if κ is weakly compact, then in the extension obtained by adding κ+ many Cohen reals, Lκω is still weakly compact. Thus IE(κ,κ,Iκ) (and hence PS+(κ,κ)) may hold without κ being inaccessible.
QUESTION. Is it consistent that PS+(κ,λ′) holds for every cardinal λ′≥κ, but κ is not inaccessible ?
DEFINITION 3.15**.**
For an infinite cardinal τ, the transversal propertyPT(κ,τ) means that for any size κ family of sets of size less than τ without a transversal (i.e. a one-to-one choice function), there exists a subfamily of size less than κ without a transversal.
FACT 3.16**.**
([27])*
It is consistent (relative to infinitely many supercompact cardinals) that PT(k,ω1) holds for every regular infinite cardinal k greater than the least fixed point of the aleph function.*
OBSERVATION 3.17**.**
Suppose that IE(τ,κ,Iκ) holds. Then so does PT(κ,τ).
Proof. Let ⟨Xα:α<κ⟩ be a sequence of sets of size less than τ with the property that for any nonzero β<κ, there is a one-to-one kβ in ∏α<βXα. Pick a one-to-one function j:⋃α<κXα→κ. For α<κ and i∈j‘‘Xα, set Aαi={γ∈κ∖α:j(kγ+1(α))=i}. There must be h∈∏α<κj‘‘Xα such that Aαh(α)∩Aβh(β)=∅ for all α,β<κ. Define g∈∏α<κXα so that j(g(α))=h(α). We will show that g is one-to-one. Thus let α<β<κ. Select γ in Aαh(α)∩Aβh(β). Then j(kγ+1(α))=h(α) and j(kγ+1(β))=h(β), which gives kγ+1(α)=g(α) and kγ+1(β)=g(β). It follows that g(α)=g(β).
□
Let PT−(κ,τ) mean that for any size κ family of sets of size less than τ with the property that any subfamily of size less than κ has a transversal, there exists a subfamily of size κ with a transversal. Then by the proof of Observation 3.17, PS+(τ,κ,τ) implies PT−(κ,τ).
QUESTION. Is it consistent that PS+(κ,κ) holds, but IE(κ,κ,Iκ) fails ?
QUESTION. What is the least possible value of κ at which PS+(κ,κ) (respectively, PS∗(κ,κ), PS(κ,κ)) may hold ?
4 Covering numbers
In this section we study the consequences of PS+ in terms of cardinal arithmetic (in the sense of Shelah).
DEFINITION 4.1**.**
Given two infinite cardinals ρ≤σ, u(ρ,σ) denotes the cofinality of the poset (Pρ(σ),⊆).
FACT 4.2**.**
(Folklore)* Let ρ≤σ be two infinite cardinals. Then σ<ρ=max{2<ρ,u(ρ,σ)}.*
DEFINITION 4.3**.**
Given four cardinals ρ1,ρ2,ρ3,ρ4 with ρ1≥ρ2≥ρ3≥ω and ρ3≥ρ4≥2, cov(ρ1,ρ2,ρ3,ρ4) denotes the least cardinality of any X⊆Pρ2(ρ1) such that for any a∈Pρ3(ρ1), there is Q∈Pρ4(X) with a⊆⋃Q.
Note that u(ρ,σ)=cov(σ,ρ,ρ,2).
FACT 4.4**.**
([30, pp. 85-86], [19])* Let ρ1,ρ2,ρ3 and ρ4 be four cardinals such that ρ1≥ρ2≥ρ3≥ω and ρ3≥ρ4≥2. Then the following hold :*
(i)
If ρ1=ρ2 and either cf(ρ1)<ρ4 or cf(ρ1)≥ρ3, then cov(ρ1,ρ2,ρ3,ρ4)=cf(ρ1).
2. (ii)
If either ρ1>ρ2, or ρ1=ρ2 and ρ4≤cf(ρ1)<ρ3, then cov(ρ1,ρ2,ρ3,ρ4)≥ρ1.
3. (iii)
If ρ3≤ρ2=cf(ρ2), ω≤ρ4=cf(ρ4) and ρ1<ρ2+ρ4, then cov(ρ1,ρ2,ρ3,ρ4)=ρ1.
9. (ix)
If ρ3=cf(ρ3), then either cf(cov(ρ1,ρ2,ρ3,ρ4))<ρ4, or cf(cov(ρ1,ρ2,ρ3,ρ4))≥ρ3.
10. (x)
Suppose that ρ3>cf(ρ2)≥ρ4 and cf(ρ3)=cf(ρ2). Then cov(ρ1,ρ2,ρ3,ρ4)=cov(ρ1,ρ,ρ3,ρ4) for some cardinal ρ with ρ2>ρ≥ρ3.
FACT 4.5**.**
(i)
([30, Remark 6.6.A p. 101])* Let χ be a singular cardinal. Then cov(χ,χ,(cf(χ))+,cf(χ))>χ+ if and only if cov(χ,χ,(cf(χ))+,2)>χ+.*
2. (ii)
([30, p. 99], [19]))* Let χ be a singular cardinal. Suppose that cov(χ,χ,(cf(χ))+,2)>χ+. Then we may find yα∈P(cf(χ))+(χ) for α<χ+ such that for any nonzero β<χ+, there is a one-to-one h∈∏α<βyα.*
Note that if f=⟨fα:α<π⟩ is an increasing, cofinal sequence in (∏A,<I), then by a result of Shelah [30, Theorem 5.4 pp. 87-88], π≤cov(supA,supA,∣A∣+,2).
OBSERVATION 4.6**.**
Let χ be a singular cardinal such that PS+((cf(χ))+,χ+) holds. Then cov(χ,χ,(cf(χ))+,2)=χ+.
Proof. Suppose otherwise. Put σ=cf(χ). By Fact 4.5, we may find yα∈Pσ+(χ) for α<χ+ such that for any nonzero β<χ+, there is a one-to-one hβ∈∏α<βyα. For α<χ+, let ⟨yαi:i<∣yα∣⟩ be a one-to-one enumeration of yα, and set Qαi={δ∈χ+∖α:hδ+1(α)=yαi}. There must be B∈[χ+]χ+ and g∈∏α∈B∣yα∣ with the property that Qαg(α)∩Qγg(γ)=∅ whenever α,γ∈B. Then clearly, the function t:χ+→χ defined by t(α)=yαg(α) is one-to-one. Contradiction.
□
Neeman [29] established the consistency relative to large cardinals of the existence of a singular strong limit cardinal χ of cofinality ω such that TP(χ+) holds and 2χ>χ+. Note that
and therefore by Observation 4.6, PS+((cf(χ))+,χ+) fails. In Neeman’s model, there is both a very good (and hence remarkably good) scale of length χ+ on χ, and a scale of length χ+ on χ that is not good (so that approachability fails, and in fact [21] there is a regular uncountable cardinal σ<χ such that Eσχ+∈/I[χ+;χ]).
To get the most out of Observation 4.6 we will vary the value of λ. We will thus be able to use the following results of pcf theory.
FACT 4.7**.**
([19])* Let σ, k, μ and ν be four infinite cardinals such that cf(σ)=σ≤k and σ<cf(μ)=μ<ν. Suppose that*
•
cov(k,ρ+,ρ+,σ)≤k++* for every cardinal ρ with σ≤ρ<min{k,μ}.*
•
cov(χ,χ,σ+,σ)=χ+* for every cardinal χ with k<χ≤ν and cf(χ)=σ.*
Then cov(ν,μ,μ,σ)≤ν+.
OBSERVATION 4.8**.**
(i)
Let π and μ be two regular cardinals such that ω≤π≤κ≤μ≤λ. Suppose that
•
either cf(λ)<π, or cf(λ)≥μ.
•
u(ρ+,κ)≤κ++* for every cardinal ρ with ω≤ρ<κ.*
•
cov(χ,χ,ω1,2)=χ+* for every cardinal χ with κ<χ<λ and cf(χ)=ω.*
Then cov(λ,μ,μ,π)=λ.
2. (ii)
Let π and μ be two regular cardinals such that ω1≤π≤κ≤μ≤λ. Suppose that
•
either cf(λ)<π, or cf(λ)≥μ.
•
cov(κ,ρ+,ρ+,ω1)≤κ++* for every cardinal ρ with ω1≤ρ<κ.*
•
cov(χ,χ,ω2,ω1)=χ+* for every cardinal χ with κ<χ<λ and cf(χ)=ω1.*
Then cov(λ,μ,μ,π)=λ.
Proof. We prove (i) and leave the similar proof of (ii) to the reader. By Fact 4.4 ((i) and (ii)), cov(τ,μ,μ,π)≥τ for every cardinal τ≥μ. Furthermore by Fact 4.7, cov(ν,μ,μ,π)≤u(μ,ν)≤ν+ for any cardinal ν with μ<ν<λ.
Case 1 : λ=μ. Then by Fact 4.4 (i),
λ≤cov(λ,μ,μ,π)≤u(μ,λ)=λ.
Case 2 : λ is the successor of some cardinal σ≥μ. Then by Fact 4.4 (i),
λ≤cov(λ,μ,μ,π)≤u(μ,λ)=max{λ,u(μ,σ)}=λ.
Case 3 : cf(λ)<π. Then by Fact 4.4 (v),
λ≤cov(λ,μ,μ,π)=sup{cov(ν,μ,μ,π):μ≤ν<λ}≤λ.
Case 4 : λ is a limit cardinal with μ≤cf(λ). Then by Fact 4.4 (vi),
which completes the proof of the claim and that of the observation.
□
5 Unbalanced partition properties
Let us start with partitions of Pκ(λ)×Pκ(λ).
DEFINITION 5.1**.**
Given two collections X and Y of subsets of Pκ(λ), an ideal J on Pκ(λ), and a cardinal ρ with 0<ρ≤κ, XY(J+,ρ)2 means that for any F:Pκ(λ)×Pκ(λ)→2 and any A∈X, there is either B∈J+∩P(A) such that {b∈B:F(a,b)=0}∈Y for all a∈B, or an increasing sequence ⟨ci:i<ρ⟩ in (A,⊂) such that F(ci,cj)=1 whenever i<j<ρ.
OBSERVATION 5.2**.**
Let J be a fine ideal on Pκ(λ). Then J+J(J+,ω)2 holds.
Proof. Fix F:Pκ(λ)×Pκ(λ)→2 and A∈J+. Define ψ:Pκ(λ)×P(A)→P(A) by
ψ(a,X)={b∈X:a⊂b and F(a,b)=1}.
Case 1 : There is Y∈J+∩P(A) such that ψ(a,Y)∈J for all a∈Y. Then clearly, F(a,b)=0 whenever a is in Y and b is in Y∖ψ(a,Y) with a⊂b.
Case 2 : For each X∈J+∩P(A), there is aX in X such that ψ(aX,X)∈J+. Inductively define Xn for n<ω by :
•
X0=A.
•
Xn+1=ψ(aXn,Xn).
Then clearly, {aXn:n<ω}⊆A. Moreover, if m<n<ω, then aXm⊂aXn and F(aXm,aXn)=1.
□
DEFINITION 5.3**.**
For an ideal J on a set X, MAD(J) (respectively, MADd(J)) denotes the collection of all Q⊆J+ such that
•
A∩B∈J (respectively, A∩B=∅) for any two distinct members A,B of Q.
•
For any C∈J+, there is A∈Q with A∩C∈J+.
Let ρ and ν be two nonzero cardinals. J is (ρ,ν)-distributive (respectively, disjointly(ρ,ν)-distributive) if given A∈J+, and Qα in MAD(J) (respectively, MADd(J)) with ∣Qα∣≤ν for α<ρ, there is B∈J+∩P(A) and h∈∏α<ρQα such that B∖h(α)∈J for every α<ρ.
OBSERVATION 5.4**.**
Suppose that J is a (θ,2)-distributive ideal on a set X, where θ is an infinite cardinal. Then J is (θ,θ)-distributive.
Proof. Let A∈J+, and Qα∈MAD(J) with ∣Qα∣≤θ for α<θ. Put Z=⋃α<θQα. There must be B∈J+∩P(A) and h∈∏W∈Z{W,X∖W} such that B∖h(W)∈J for every W∈Z. Now given α<θ, we have Qα∈MAD(J), so there is a (unique) W∈Qα such that h(W)=W.
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OBSERVATION 5.5**.**
Suppose that J+J(J+,σ+)2 holds, where J is a κ-complete, fine ideal on Pκ(λ), and σ is an infinite cardinal. Then J is disjointly (σ,λ<κ)-distributive.
Proof. Let A∈J+, and Qα in MADd(J) for α<σ. For α<σ, put Xα0=Pκ(λ)∖⋃Qα, and let ⟨Xαi:0<i<∣Qα∣⟩ be a one-to-one enumeration of Qα. Define g:σ×Pκ(λ)→λ<κ so that for any a∈Pκ(λ) and any α<σ, a∈Xαg(α,a). Now define F:Pκ(λ)×Pκ(λ)→2 by : F(a,b)=1 if and only if there is α<σ such that g(α,a)=g(α,b), and for the least such α, g(α,a)>g(α,b).
Case 1 : There is B∈J+, and Za∈J for a∈B such that F(a,b)=0 whenever a,b∈B and b∈/Za. Assume toward a contradiction that there is γ<σ such that {B∖Xγi:0<i<∣Qγ∣}⊆J+, and let α denote the least such γ. Let h∈∏β<α{k:0<k<∣Qβ∣} such that {B∖Xβh(β):β<α}⊆J. There must be 0<i<j<∣Qα∣ such that {B∩Xαi,B∩Xαj}⊆J+. Pick a in (B∩Xαj)∖⋃β<α(B∖Xβh(β)), and b in (B∩Xαi)∖(Za∪⋃β<α(B∖Xβh(β))). Then F(a,b)=1. Contradiction.
Case 2 : There is an increasing sequence ⟨aδ:δ<σ+⟩ in (A,⊂) such that F(aγ,aδ)=1 whenever γ<δ<σ+. We will show that this is contradictory. For this we inductively define ξα<σ+ for α<σ so that g(α,aξα)=g(α,aδ) whenever ξα<δ<σ+. Thus suppose that ξβ has been defined for each β<α.
Claim. There is ξ<σ+ such that g(α,aξ)=g(α,aδ) whenever ξ<δ<σ+.
Proof of the claim. Suppose otherwise. Inductively define γn for n<ω so that
•
γ0=sup{ξβ:β<α}.
•
γn+1>γn and g(α,aγn+1)=g(α,aγn).
Then g(α,aγ0)>g(α,aγ1)>g(α,aγ2)⋯. This contradiction completes the proof of the claim.
Using the claim, we let ξα= the least ξ<σ+ such that g(α,aξ)=g(α,aδ) whenever ξ<δ<σ+.
Finally pick γ,δ so that sup{ξη:η<σ}≤γ<δ<σ+. Then g(η,aξ)=g(η,aδ) for all η<σ. Hence F(aγ,aδ)=0, which yields the desired contradiction.
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The remainder of the section is devoted to partitions of κ×Pκ(λ).
DEFINITION 5.6**.**
Given two collections X and Y of subsets of Pκ(λ), an ideal J on Pκ(λ), and a cardinal ρ with 0<ρ≤κ, XYκ(J+,ρ)2 means that for any F:κ×Pκ(λ)→2 and any A∈X, there is either B∈J+∩P(A) such that {b∈B:F(sup(a∩κ),b)=0}∈Y for all a∈B, or an increasing sequence ⟨ci:i<ρ⟩ in (A,⊂) such that F(sup(ci∩κ),cj)=1 whenever i<j<ρ.
OBSERVATION 5.7**.**
XJ(J+,ρ)2* implies XJκ(J+,ρ)2.*
OBSERVATION 5.8**.**
Assuming J is fine, the following are equivalent :
(i)
XJκ(J+,ρ)2* holds.*
2. (ii)
For any G:κ×Pκ(λ)→2 and any A∈X, there is either B∈J+∩P(A) such that {b∈B:G(sup(a∩κ),b)=0}∈J for all a∈B, or an increasing sequence ⟨ci:i<ρ⟩ in (A,⊂) such that sup(ci∩κ)<sup(cj∩κ) and G(sup(ci∩κ),cj)=1 whenever i<j<ρ.
Proof. (i) → (ii) : Given G:κ×Pκ(λ)→2 and A∈X, define F:κ×Pκ(λ)→2 by : F(α,b)=1 if and only if G(α,b)=1 and α<sup(a∩κ).
(ii) → (i) : Trivial.
□
DEFINITION 5.9**.**
An ideal J on Pκ(λ) is κ-normal if for any A∈J+ and any f:A→κ with the property that f(a)∈a for all a∈A, there is B∈J+∩P(A) such that f is constant on B.
We let NSκ,λκ denote the smallest κ-normal, fine ideal on Pκ(λ).
DEFINITION 5.10**.**
We let Ωκ,λ denote the set of all a∈Pκ(λ) such that a∩κ is an infinite limit ordinal.
An ideal J on Pκ(λ) is a weak π-point if for any A∈J+ and any f:κ→J, there is B∈J+∩P(A) such that B∩f(α)∈Iκ,λ for every α∈κ.
J is a weak χ-point if for any A∈J+ and any g:κ→Pκ(λ), there is B∈J+∩P(A) such that g(sup(a∩κ)⊆b for all a,b∈B with sup(a∩κ)<sup(b∩κ).
OBSERVATION 5.13**.**
(i)
Iκ,λ* is a weak π-point.*
2. (ii)
Any weak χ-point is fine.
3. (iii)
If J is fine and κ-normal, then it is both a weak π-point and a weak χ-point.
DEFINITION 5.14**.**
Given a collection X of subsets of Pκ(λ), an ideal J on Pκ(λ), and a cardinal ρ with 0<ρ≤κ, X≺κ(J+,ρ)2 means that for any F:κ×Pκ(λ)→2 and any A∈X, there is either B∈J+∩P(A) such that F(sup(a∩κ),b)=0 whenever a,b∈B are such that a⊂b and sup(a∩κ)<sup(b∩κ), or an increasing sequence ⟨ci:i<ρ⟩ in (A,⊂) such that sup(ci∩κ)<sup(cj∩κ) and F(sup(ci∩κ),cj)=1 whenever i<j<ρ.
FACT 5.15**.**
([16])* Let ρ be an uncountable cardinal less than or equal to κ, and X={a∈Ωκ,λ:cf(a∩κ)<ρ}. Then X≺κ(J+,ρ)2 fails.*
OBSERVATION 5.16**.**
Suppose that J+Jκ(J+,ρ)2 holds, where J is both a weak π-point and a weak χ-point, and ρ is a cardinal with 0<ρ≤κ. Then J+≺κ(J+,ρ)2 holds.
Proof. Use Observation 5.8.
□
OBSERVATION 5.17**.**
Suppose that J+Jκ(J+,ρ)2 holds, where J is a κ-normal, fine ideal on Pκ(λ), and ρ is a cardinal with ω<ρ<κ. Then {a∈Ωκ,λ:cf(a∩κ)<ρ}∈J.
Proof. By Fact 5.15 and Observations 5.13 and 5.16.
□
Let us now concentrate on J+Jκ(J+,κ)2.
OBSERVATION 5.18**.**
Suppose that {Pκ(λ)}Jκ(J+,κ)2 holds, where J is a κ-complete, fine ideal on Pκ(λ). Then κ is weakly compact.
Proof. Given f:κ×κ→2, define F:κ×Pκ(λ)→2 by : F(α,b)=1 if and only if α<sup(b∩κ) and f(α,sup(b∩κ))=1.
Case 1 : There is B∈J+ such that Ta∈J for all a∈B, where Ta={b∈B:F(sup(a∩κ),b)=0}. Proceeding by induction, define aα∈A for α<κ so that for any β<α, aα∈/Taβ and sup(aβ∩κ)<sup(aα∩κ). Then clearly, f(sup(aβ∩κ),sup(aα∩κ))=0 whenever β<α<κ.
Case 2 : There is an increasing sequence ⟨ci:i<κ⟩ in (A,⊂) such that F(sup(ci∩κ),cj)=1 whenever i<j<κ. Then given i<j<κ, we have sup(ci∩κ)<sup(cj∩κ) and f(sup(ci∩κ),sup(cj∩κ))=1.
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DEFINITION 5.19**.**
Given an ideal J on Pκ(λ), we let J↾κ denote the set of all X⊆κ such that {a∈Pκ(λ):sup(a∩κ)∈X} lies in J.
DEFINITION 5.20**.**
An ideal J on κ is weakly selective if for any A∈J+ and any partition Q of A into sets in J, there is B∈J+∩P(A) such that ∣B∩W∣≤1 for every W∈Q.
OBSERVATION 5.21**.**
(i)
J↾κ* is an ideal on κ.*
2. (ii)
If J is fine, then so is J↾κ.
3. (iii)
If J is κ-complete, then so is J↾κ.
4. (iv)
If J is both a weak π-point and a weak χ-point, then J↾κ is weakly selective.
5. (v)
If J is fine and κ-normal, then J↾κ is normal.
6. (vi)
J↾κ⊆K↾κ* for every ideal K on Pκ(λ) extending J.*
Given an ideal J on a set X, we let cof(J) denote the least size of any B⊆J such that J=⋃B∈BP(B).
Assuming that J is κ-complete, but not κ+-complete, we let cof(J) denote the least size of any B⊆J such that for any A∈J, there is b∈Pκ(B) with A⊆⋃b.
DEFINITION 5.24**.**
The dominating numberdκ denotes the least size of any F⊆κκ with the property that for any g∈κκ, there is f∈F such that g(α)<f(α) for every α∈κ.
We let dκ denote the least size of any F⊆κκ with the property that for any g∈κκ, there is x∈Pκ(F) such that g(α)<sup{f(α):f∈x} for every α∈κ.
Let J be a κ-complete, fine ideal on κ such that cof(J)≤λ. Then J=(Iκ,λ∣A)↾κ for some A∈Iκ,λ+∩P(Ωκ,λ).
2. (ii)
Suppose dκ≤λ. Then NSκ=(Iκ,λ∣C)↾κ for some C∈NSκ,λ∗.
3. (iii)
Let J be a κ-complete, fine ideal on κ such that cof(J)≤λ. Then J=(Iκ,λ∣A)↾κ for some A∈Iκ,λ+ such that (A,⊂) and (Pκ(λ),⊂) are isomorphic.
Proof. (i) : Pick B⊆J with ∣B∣=cof(J) such that for any B∈J, there is b∈Pκ(B) with B⊆⋃b. Select v⊆λ∖κ with ∣v∣=∣B∣, and a bijection h:v→B. Let A be the set of all a∈Ωκ,λ such that a∩κ∈/h(α) for all α∈v∩a.
Claim. Let X∈J+. Then {a∈A:a∩κ∈X}∈Iκ,λ+.
Proof of the claim. Fix c∈Pκ(λ). Pick δ∈X∖(⋃α∈v∩ch(α)) with c∩κ⊆δ. Set a=δ∪(c∖κ). Then clearly, v∩c=v∩a. Moreover, c⊆a and a∈A, which completes the proof of the claim.
By the claim, A∈Iκ,λ+, and moreover (Iκ,λ∣A)↾κ⊆J. For the reverse inclusion, fix X∈J. We may find e∈Pκ(v) such that X⊆⋃α∈eh(α). Then a∩κ∈/X for all a∈A with e⊆a. Hence X∈(Iκ,λ∣A)↾κ.
(ii) : By Fact 5.11, 5.22 and 5.25 and the proof of (i).
(iii) : Let B, v and h be as in the proof of (i), and let A be the set of all a∈Pκ(λ) such that
•
sup(a∩κ)∈a.
•
sup(a∩κ)∈/h(α) for all α∈v∩a.
Claim. Let X∈J+. Then {a∈A:sup(a∩κ)∈X}∈Iκ,λ+.
Proof of the claim. Given c∈Pκ(λ), pick δ∈X∖(⋃α∈v∩ch(α)) with sup(c∩κ)<δ. Set a=c∪{δ}. Then clearly, c⊆a, a∈A and sup(a∩κ)=δ, which completes the proof of the claim.
By the claim, A∈Iκ,λ+, and moreover (Iκ,λ∣A)↾κ⊆J. For the reverse inclusion, proceed as in the proof of (i). Finally, define φ:Pκ(λ)→κ and ψ:Pκ(λ)→Pκ(λ) by φ(a)= the least δ>sup(a∩κ) such that δ∈/⋃α∈v∩ah(α), and ψ(a)=a∪{φ(a)}. It is easy to see that ψ is an isomorphism from (Pκ(λ),⊂) onto (A,⊂).
□
Notice that if (A,⊂) and (Pκ(λ),⊂) are isomorphic and, say, {Pκ(λ)}Iκ,λκ(Iκ,λ+,κ)2 holds, then so does {A}Iκ,λκ(Iκ,λ+,κ)2.
DEFINITION 5.27**.**
Given a collection W of subsets of κ, an ideal K on κ, and a cardinal ρ with 0<ρ≤κ, W→(K+,ρ)2 means that for any f:κ×κ→2 and any A∈W, there is either B∈K+∩P(A) such that f(α,β)=0 for any α<β in B, or an increasing sequence ⟨γi:i<ρ⟩ in (A,<) such that f(γi,γj)=1 whenever i<j<ρ.
PROPOSITION 5.28**.**
Suppose that J+Jκ(J+,κ)2 holds, where J is both a weak π-point and a weak χ-point. Then (J↾κ)+→((J↾κ)+,κ)2 holds.
Proof. Let X∈(J↾κ)+ and f:κ×κ→κ be given. Put A={a∈Pκ(λ):sup(a∩κ)∈X}, and define F:κ×A→2 by : F(α,b)=1 if and only if α<sup(b∩κ) and f(α,sup(b∩κ))=1.
Case 1 : There is B∈J+∩P(A), and Za∈J for a∈B such that F(sup(a∩κ),b)=0 whenever a,b∈B and b∈/Za. Set T={sup(a∩κ):a∈B}. For α∈T, pick aα∈B with sup(aα∩κ)=α. There must be S∈J+∩P(B) such that for any α∈T, and any a,b∈S with sup(a∩κ)=α<sup(b∩κ), b∈/Zaα. Now put Y={sup(b∩κ):b∈S}. Then Y∈(J↾κ)+∩P(X). Given α<β in Y, we may find a,b∈S such that sup(a∩κ)=α and sup(b∩κ)=β. Then b∈/Zaα. Since aα,b∈B, it follows that
0=F(sup(aα∩κ),b)=f(α,sup(b∩κ))=f(α,β).
Case 2 : There is an increasing sequence ⟨aδ:δ<κ⟩ in (A,⊂) such that F(sup(aγ∩κ),aδ)=1 whenever γ<δ<κ. Then clearly by definition of F, given γ<δ<κ, we have sup(aγ∩κ)<sup(aδ∩κ), and moreover f(sup(aγ∩κ),sup(aδ∩κ))=1.
□
DEFINITION 5.29**.**
We let NWCκ denote the set of all A⊆κ such that {A}→(NSκ+,κ)2 does not hold.
FACT 5.30**.**
(i)
([3])* Assuming κ is weakly compact, NWCκ is the smallest normal ideal K on κ such that K+→(K+,κ)2.*
2. (ii)
([3])* The set of all those cardinals less than κ that are not inaccessible belongs to NWCκ.*
3. (iii)
([16])* Let J be a κ-normal, fine ideal on Pκ(λ). Then J+Jκ(J+,κ)2 holds if and only if J↾κ extends NWCκ.*
Notice that by Shelah’s Revised GCH theorem [31, Conclusion 1.2], for any uncountable strong limit cardinal τ, there is θ<τ with the property that if θ≤κ<τ<λ, then cov(λ,κ+,κ+,κ)=λ.
PROPOSITION 5.32**.**
Suppose dκ≤λ=cov(λ,κ+,κ+,κ). Then the following hold.
(i)
Let σ be an infinite cardinal. Then for any D∈NSκ,λ∗, J+Jκ(J+,σ+)2 does not hold, where J=Iκ,λ∣{a∈D∩Ωκ,λ:cf(a∩κ)≤σ}.
2. (ii)
For any D∈NSκ,λ∗ and any X∈NSκ+∩NWCκ, J+Jκ(J+,κ)2 does not hold, where J=Iκ,λ∣{a∈D:sup(a∩κ)∈X}.
Proof. (ii) : Assume toward a contradiction that J+Jκ(J+,κ)2 holds. By Fact 5.31, there is C∈NSκ,λ∗ such that Iκ,λ∣C is κ-normal. Put K=J∣(C∩Ωκ,λ). Then clearly, K is κ-normal and K+Kκ(K+,κ)2 holds. Hence by Observation 5.21 and Proposition 5.28, K↾κ is a normal, fine ideal on κ such that (K↾κ)+→((K↾κ)+,κ)2 holds. By Fact 5.30, it follows that X∈K↾κ. Contradiction.
(i) : Proceed as in the proof of (ii), but this time appeal to Observation 5.17.
□
6 Balanced partition properties
We now turn to stronger partition properties that (in the case when cf(λ)=κ) will imply that cov(λ,κ+,κ+,κ)=λ.
DEFINITION 6.1**.**
Given 2≤n<ω, two collections X and Y of subsets of Pκ(λ), an ideal J on Pκ(λ), and a cardinal ρ, XY(J+)ρn means that for any F:(Pκ(λ))n→ρ and any A∈X, there is i<ρ, B∈J+∩P(A), and Za∈Y for a∈B such that F(a1,a2,⋯,an)=i whenever a1,a2,⋯,an∈B and am+1∈/Za1∪Za2∪⋯∪Zam for 1≤m<n.
XYκ(J+)ρ2 means that for any F:κ×Pκ(λ)→ρ and any A∈X, there is i<ρ and B∈J+∩P(A) such that {b∈B:F(sup(a∩κ),b)=i}∈Y for all a∈B.
OBSERVATION 6.2**.**
(i)
Suppose that J+J(J+)22 holds. Then J+J(J+)n2 holds for every n with 0<n<ω.
2. (ii)
XJ(J+)ρ2* implies XJκ(J+)ρ2.*
3. (iii)
Suppose that J is fine and (κ,2)-distributive. Then J+Jκ(J+)ρ2 whenever 0<ρ<κ.
OBSERVATION 6.3**.**
Suppose that {Pκ(λ)}J(J+)ρ2 holds, where ρ is an infinite cardinal and J is a fine ideal on Pκ(λ). Then PS+(ρ+,κ,ν) holds for any cardinal ν with κ≤ν≤λ.
Proof. Let ν be a cardinal with κ≤ν≤λ, and for x∈Pκ(ν), Qx be a partition of Pκ(ν) with ∣Qx∣≤ρ. For x∈Pκ(ν), let ⟨Qxξ:ξ<∣Qx∣⟩ be a one-to-one enumeration of Qx. Define F:Pκ(λ)×Pκ(λ)→ρ by : F(a,c)=ξ if and only if c∩ν∈Qa∩νξ. There must be ξ<ρ and B∈J+ such that Ta∈J for all a∈B, where Ta={c∈B:F(a,c)=ξ}. Set X={a∩ν:a∈B}. Notice that X∈Iκ,ν+. Given x,y∈X, select a,b∈B such that x=a∩ν and y=b∩ν. Pick c∈B∖(Ta∪Tb) such that a∪b⊆c. Then obviously, x∪y⊆c∩ν. Moreover, c∩ν∈Qxξ∩Qyξ.
□
OBSERVATION 6.4**.**
Suppose that {Pκ(λ)}J(J+)ω2 holds, where J is a κ-complete, fine ideal on Pκ(λ). Then the following hold :
(i)
Assume cf(λ)=κ. Then cov(λ,κ+,κ+,κ)=λ.
2. (ii)
Let τ be a cardinal with κ+≤τ<λ. Then u(κ+,τ) equals τ if cf(τ)>κ, and τ+ otherwise.
Proof. For any cardinal ν such that κ<ν<λ and cf(ν)=ω, we have that PS+(ω1,ν+) holds by Proposition 3.5 and Observation 6.3, and hence that cov(ν,ν,ω1,2)=ν+ by Observation 4.6. Furthermore by Observation 5.18, κ is inaccessible. Now for (i), apply Observation 4.8 (i). To obtain (ii), appeal to Facts 4.4 and 4.7 if cf(τ)≤κ, and to Observation 4.8 (i) otherwise.
□
The remainder of the section is, just like the end of the previous section, devoted to negative partition relations.
DEFINITION 6.5**.**
Given two collections X and Y of subsets of Pκ(λ), an ideal J on Pκ(λ) and two nonzero cardinals σ and ρ, the square bracket partition relationXYσ[J+]ρ2 means that for any F:σ×Pκ(λ)→ρ and any A∈X, there is B∈J+∩P(A) and ξ∈ρ such that {b∈B:F(sup(a∩σ),b)=ξ}∈Y for all a∈B.
DEFINITION 6.6**.**
For any cardinal χ with κ≤χ≤λ, we let Sκ,λχ={a∈Pκ(λ):∣a∩χ∣=∣a∩κ∣}.
Suppose that κ is weakly inaccessible and 2κ≤λ=cov(λ,κ+,κ+,κ). Then {C∩Sκ,λλ}Iκ,λκ[Iκ,λ+]λ2 fails for some C∈NSκ,λ∗.
2. (ii)
Suppose that κ is weakly Mahlo and dκ≤λ, and let A be the set of all a∈Sκ,λdκ such that a∩κ is a regular infinite cardinal. Then there is D∈NSκ,λ∗ and F:κ×Pκ(λ)→λ with the property that for any B∈(NSκ,λκ)+∩P(A∩D) and any ξ∈λ, one may find a,b∈B with a∩κ<b∩κ and F(a∩κ,b)=ξ.
OBSERVATION 6.8**.**
Suppose that κ is weakly Mahlo, dκ≤λ, and J is a κ-normal, fine ideal on Pκ(λ) such that J+Jκ[J+]λ2 holds. Then A∩C∈J for some C∈NSκ,λ∗, where A denotes the set of all a∈Sκ,λdκ such that a∩κ is a regular infinite cardinal.
Proof. Suppose otherwise. Then clearly, A∈NSκ,λ+. Now let D∈NSκ,λ∗ and F:κ×Pκ(λ)→λ. Since {A∩D}Jκ[J+]λ2 and J is κ-normal, there must be B∈J+∩P(A∩D∩Ωκ,λ) and ξ∈λ such that F(a∩κ,b)=ξ whenever a,b∈B are such that a∩κ∈b. This contradicts Fact 6.7.
□
COROLLARY 6.9**.**
Suppose that dκ≤λ and J+Jκ(J+)22 holds, where J is a κ-normal, fine ideal on Pκ(λ). Then Sκ,λdκ∩C∈J for some C∈NSκ,λ∗.
Proof. By Observation 5.18 and Fact 5.30, κ is weakly compact, and moreover the set of all a∈Pκ(λ) such that a∩κ is an inaccessible cardinal lies in J∗.
□
COROLLARY 6.10**.**
Suppose that κ is weakly Mahlo and dκ≤λ=cov(λ,κ+,κ+,κ). Then for any D∈NSκ,λ∗, J+Jκ[J+]λ2 does not hold, where J=Iκ,λ∣(D∩Sκ,λdκ).
Proof. Assume toward a contradiction that there exists D∈NSκ,λ∗ such that J+Jκ[J+]λ2 holds, where J=Iκ,λ∣(D∩Sκ,λdκ). By Fact 5.31, we may find C∈NSκ,λ∗ such that Iκ,λ∣C is κ-normal. Put K=J∣C. Then clearly, K is κ-normal, and moreover K+Kκ(K+)22 holds. This contradicts Observation 6.8.
□
A similar result will be obtained as a variant of Proposition 5.32 (ii).
PROPOSITION 6.11**.**
Suppose that J+Jκ(J+)22 holds, where J is a κ-normal, fine ideal on Pκ(λ). Then (J↾κ)+→((J↾κ)+)2 holds.
Proof. By the proof of Proposition 5.28.
□
OBSERVATION 6.12**.**
Given a κ-complete, fine ideal J on κ, the following are equivalent:
(i)
J* is (κ,2)-distributive.*
2. (ii)
Let 0<η≤κ, and Qα∈MADd(J) for α<η be such that Qβ⊆⋃W∈QαP(W) whenever α<β<η. Then there is B∈J+∩P(A) and h∈∏α<ηQα such that B∖h(α)∈J for all α<κ.
Proof. (i) → (ii) : By Observation 5.4.
(ii) → (iii) : Suppose that (ii) holds. We claim that κ is inaccessible. Suppose otherwise, and let ν be the least cardinal such that 2ν≥κ.
Let ⟨Xξ:ξ<κ⟩ be a sequence of pairwise distinct subsets of ν. For δ<ν, put Aδ0={ξ<κ:δ∈Xξ} and Aδ1=κ∖Aδ0. Now let Q0={κ}, and for 0<α<ν, Qα={⋂δ<αAδk(δ):k∈α2}∩J+. Then ∣⋂α<νh(α)∣≤1 for all h∈∏α<νQα, which yields the desired contradiction.
Now suppose that A∈J+ and for α<κ, Wα∈MADd(J) with ∣Wα∣≤2. For α<κ, let
Tα={⋂β≤αg(β):g∈∏β≤αWβ}∩J+.
There must be B∈J+∩P(A) and f∈∏α<κTα such that B∖f(α)∈J for all α<κ. For α<κ, let gα∈∏β≤αWβ be such that f(α)=⋂β≤αgα(β). Then it is easy to see that ⋃α<κgα∈∏β≤κWβ. Moreover, B∖(⋃α<κgα)(β)∈J for all β<κ.
□
DEFINITION 6.13**.**
Given an ideal J on κ, 2≤n<ω, a collection X of subsets of κ and an ordinal ρ, X→(J+)ρn means that for any F:κn→ρ and any A∈X, there is i<ρ and B∈J+∩P(A) such that F(α1,α2,⋯,αn)=i whenever α1<a2<⋯<αn are in B.
FACT 6.14**.**
Given a κ-complete, fine ideal J on κ, the following are equivalent :
(i)
J+→(J+)22.
2. (ii)
J+→(J+)ρn* whenever 0<n<ω and 0<ρ<κ.*
3. (iii)
J* is (κ,2)-distributive and weakly selective.*
Proof. By Theorem 9 in [12] and Observation 6.12.
□
DEFINITION 6.15**.**
κ is completely ineffable if there exists a normal, (κ,2)-distributive, fine ideal on κ.
FACT 6.16**.**
Suppose that κ is completely ineffable. Then there exists a smallest normal, (κ,2)-distributive, fine ideal on κ.
Proof. By the proof of Corollary 3 in [12] and Observation 6.12.
□
DEFINITION 6.17**.**
We let NCIκ denote the smallest normal, (κ,2)-distributive, fine ideal on κ if κ is completely ineffable, and P(κ) otherwise.
PROPOSITION 6.18**.**
Suppose dκ≤λ=cov(λ,κ+,κ+,κ). Then for any D∈NSκ,λ∗ and any X∈NSκ+∩NCIκ, J+Jκ(J+)22 does not hold, where J=Iκ,λ∣{a∈D:sup(a∩κ)∈X}.
Proof. Assume toward a contradiction that there are D∈NSκ,λ∗ and X∈NSκ+∩NCIκ such that J+Jκ(J+)22 holds, where J=Iκ,λ∣{a∈D:sup(a∩κ)∈X}. By Fact 5.31, there is C∈NSκ,λ∗ such that Iκ,λ∣C is κ-normal. Put K=J∣C. Then clearly, K is κ-normal, and moreover K+Kκ(K+)22 holds. Hence by Observation 5.21, Proposition 6.11 and Fact 6.14, K↾κ is a normal, (κ,2)-distributive, fine ideal on κ. It follows that X∈J↾κ. Contradiction.
□
PROPOSITION 6.19**.**
Suppose that dκ≤λ and cfλ)=κ. Then the following hold :
(i)
For any D∈NSκ,λ∗, J+J(J+)ω2 does not hold, where J=Iκ,λ∣(D∩Sκ,λdκ).
2. (ii)
For any D∈NSκ,λ∗ and any X∈NSκ+∩NCIκ, J+J(J+)ω2 does not hold, where J=Iκ,λ∣{a∈D:sup(a∩κ)∈X}.
Proof. By Observation 6.4, Corollary 6.10 and Proposition 6.18.
□
In contrast to this, by a result of Usuba (see the proof of Theorem 1.9 in [36]), it is consistent relative to a large cardinal that \sayκ is not subtle, but Iκ,λ+→(Iκ,λ+)ηn holds for any n<ω and any η<κ.
7 Mild ineffability
DEFINITION 7.1**.**
κ is mildly λ-ineffable if, given sa⊆a for a∈Pκ(λ), there exists S⊆λ with the property that for any b∈Pκ(λ), there is a∈Pκ(λ) such that b⊆a and S∩b=sa∩b.
We will establish that if κ is mildly λ-ineffable and cf(λ)=κ, then cov(λ,κ+,κ+,κ)=λ. We need some preparation.
Given Wα⊆Pκ(λ) for α<λ, there is h∈∏α<λ{Wα,Pκ(λ)∖Wα} such that ⋂α∈eh(α)∈Iκ,λ+ for every nonempty e∈Pκ(λ).
OBSERVATION 7.5**.**
Suppose that κ is mildly λ-ineffable, and for each α<λ, let Qα be a partition of Pκ(λ) into less than κ many pieces. Then there is h∈∏α<λQα such that ⋂α∈eh(α)∈Iκ,λ+ for every nonempty e∈Pκ(λ).
Proof. By Fact 7.4, we may find h∈∏W∈⋃α<λQα{W,Pκ(λ)∖W} such that ⋂W∈xh(W)∈Iκ,λ+ for every nonempty x∈Pκ(⋃α<λQα). Now given α<λ, we have ⋂W∈Qα(Pκ(λ)∖W)=∅, and consequently Qα∩ran(h)=∅.
□
TP(κ,λ) may be reformulated in the same way.
PROPOSITION 7.6**.**
The following are equivalent :
(i)
TP(κ,λ).
2. (ii)
For each α<λ, let Qα be a partition of {x∈Pκ(λ):α∈x} into less than κ many pieces. Suppose that
∣{⋂α∈dg(α):g∈∏α∈dQα}∣<κ
for any nonempty d∈Pκ(λ). Then there is h∈∏α<λQα such that ⋂α∈eh(α)∈Iκ,λ+ for every nonempty e∈Pκ(λ).
3. (iii)
For each α<λ, let Qα be a partition of {x∈Pκ(λ):α∈x} with ∣Qα∣≤2. Suppose that
∣{⋂α∈dg(α):g∈∏α∈dQα}∣<κ
for any nonempty d∈Pκ(λ). Then there is h∈∏α<λQα such that ⋂α∈eh(α)∈Iκ,λ+ for every nonempty e∈Pκ(λ).
Proof. (i) → (ii) : Assume that TP(κ,λ) holds. Let Qα be a partition of {x∈Pκ(λ):α∈x} into less than κ many pieces for α<κ such that
∣{⋂α∈dg(α):g∈∏α∈dQα}∣<κ
for any nonempty d∈Pκ(λ). For α<λ, let ⟨Qαξ:ξ<∣Qα∣⟩ be a one-to-one enumeration of Qα. Select a bijection f:κ×λ→λ. For a∈Pκ(λ), define ta:a→2 as follows. Given ξ<κ and α<λ such that f(ξ,α)∈a, we let ta(j(ξ,α))=1 just in case α∈a and a∈Qαξ. For c∈Pκ(λ), let Ac denote the collection of all α<λ such that f(ξ,α)∈c for some ξ<κ. Let C be the set of all c∈Pκ(λ) such that Ac⊆c. Note that C∈NSκ,λ∗.
Claim 1. Let c∈C. Then ∣{ta∣c:c⊆a}∣<κ.
Proof of Claim 1. Suppose otherwise, and let ai∈Pκ(λ) for i<κ be such that
•
c⊆ai for all i<κ.
•
tai∣c=taj∣c whenever i<j<κ.
For i<κ, define ki:Ac→κ so that ai∈Qαki(α) for all α∈Ac. Now given i<j<κ, we may find α∈Ac and ξ<κ such that f(ξ,α)∈c and tai(f(ξ,α))=tai(f(ξ,α)). Then it is easy to see that ki(α)=kj(α). Hence, ki=kj. This contradiction completes the proof of the claim.
By Claim 1, we may find T:λ→2 such that for any v∈Pκ(λ), there is a∈Pκ(λ) with v⊆a and T∣v=ta∣v.
Claim 2. Let α<λ. Then T(f(ξ,α))=1 for some ξ<∣Qα∣.
Proof of Claim 2. Suppose otherwise. Pick v∈Pκ(λ) such that {α}∪{f(ξ,α):ξ<∣Qα∣}⊆v. There must be a∈Pκ(λ) such that v⊆a and T∣v=ta∣v. Then a∈/Qαξ for all ξ<∣Qα∣. This contradiction completes the proof of the claim.
Claim 3. Let α<λ. Then ∣{ξ<λ:T(f(ξ,α))=1}≤1.
Proof of Claim 3. Let ξ1,ξ2<λ be such that T(f(ξ1,α))=T(f(ξ2,α))=1. Pick v∈Pκ(λ) such that {α,f(ξ1,α),f(ξ2,α)}⊆v. There must be a∈Pκ(λ) such that v⊆a and T∣v=ta∣v. Then a∈Qαξ1∩Qαξ2. Hence ξ1=ξ2, which completes the proof of the claim.
Using Claims 2 and 3, define H∈∏α<λ∣Qα∣ by H(α)= the unique ξ<∣Qα∣ such that T(f(ξ,α))=1. Now given e,w∈Pκ(λ)∖{∅}, set v=e∪w∪{f(H(α),α):α∈e}. We may find a∈Pκ(λ) such that v⊆a and T∣v=ta∣v. Then clearly,
•
w⊆a.
•
a∈⋂α∈eQαH(α).
(ii) → (iii) : Trivial.
(iii) → (i) : Assume that (iii) holds. Let ta:a→2 for a∈Pκ(λ) be such that ∣{ta∣d:d⊆a}∣<κ for all d∈Pκ(λ). For α<λ and i<2, let Qαi be the set of all a∈Pκ(λ) such that α∈a and ta(α)=i.
Claim. Let d∈Pκ(λ)∖{∅}. Then
∣{⋂α∈du(α):u∈∏α∈d{Qα0,Qα1}}∣<κ
.
Proof of the claim. Suppose otherwise. Pick gξ:d→2 for ξ<κ so that ⋂α∈dQαgη(α)=⋂α∈dQαgξ(α) whenever η<ξ<κ. For ξ<κ with ⋂α∈dQαgξ(α)=∅, pick aξ∈⋂α∈dQαgξ(α). Notice that taξ∣d=gξ. Thus ∣{taξ∣d:ξ<κ}∣=κ. This contradiction completes the proof of the claim.
By the claim, we may find T:λ→2 such that ⋂α∈eQαT(α)∈Iκ,λ+ for every nonempty e∈Pκ(λ). It remains to observe that for any e∈Pκ(λ)∖{∅} and any a∈⋂α∈eQαT(α), we have ta∣e=T∣e.
□
DEFINITION 7.7**.**
Given a set P and a κ-complete ideal J on P, we denote by IEκ2(J) the following statement : Suppose that for each p∈P, there is a partition Qp of P with ∣Qp∣<κ. Then there is h∈∏p∈PQp and a κ-complete ideal K on P extending J such that ran(h)⊆K∗.
OBSERVATION 7.8**.**
Suppose that κ is mildly λ-ineffable and λ is regular. Then IEκ2(Iλ) holds.
Proof. For each α<λ, let Wα be a partition of λ with ∣Wα∣<κ. For α<λ, put
Qα={{a∈Pκ(λ):supa∈T}:T∈Wα}.
By Observation 7.5, we may find h∈∏α<λWα such that
{a∈Pκ(λ):supa∈⋂α∈eh(α)}∈Iκ,λ+
for every nonempty e∈Pκ(λ). It is easy to see that ⋂α∈eh(α)∈Iλ+ for all e∈Pκ(λ)∖{∅}.
□
Note that if λ is weakly compact, then by Fact 7.2 and Observation 7.8, IEκ2(Iλ) (in fact IEλ2(Iλ)) holds. Thus the converse of Observation 7.8 does not hold.
OBSERVATION 7.9**.**
Suppose that κ is mildly λ-ineffable. Then cov(ν,ν,(cf(ν)+,2)=ν+ for each singular cardinal ν with κ<ν<λ.
Proof. Given a singular cardinal ν with κ<ν<λ, κ is mildly ν+-ineffable by Fact 7.2, so PS+((cf(ν)+,ν+) holds by Observation 7.7, and therefore cov(ν,ν,(cf(ν))+,2)=ν+ by Observation 4.6.
□
FACT 7.10**.**
([35])* Suppose that cf(λ)≥κ and κ is mildly λ-ineffable. Then λ<κ=λ.*
Proof. Since κ is inaccessible by Fact 7.2, it follows from Fact 4.2 and Observations 4.8 (i) and 7.9 that λ<κ=u(κ,λ)=λ.
□
Usuba [35] asked whether λ<κ=λ+ whenever κ is mildly λ-ineffable and cf(λ)<κ. The following provides a partial answer to this question.
PROPOSITION 7.11**.**
(i)
Suppose that ω<cf(λ)<κ, and κ is mildly ν-ineffable for every cardinal ν with κ≤ν<λ. Then λ<κ=λ+.
2. (ii)
Suppose that ω=cf(λ), and κ is mildly ν-ineffable for every cardinal ν with κ≤ν<λ. Then λ<κ=cov(λ,λ,ω1,2).
Proof. (i) : κ is inaccessible by Fact 7.2, so by Facts 4.2 and 4.7 and Observation 7.9, λ+≤λ<κ=u(κ,λ)≤λ+.
(ii) : Use Observation 4.10.
□
PROPOSITION 7.12**.**
Suppose that κ is mildly λ-ineffable and cf(λ)=κ. Then cov(λ,κ+,κ+,κ)=λ.
Proof. Since κ is inaccessible by Fact 7.2, the result follows from Observations 7.9 and 4.8 (use (i) if cf(λ)=ω, and (ii) otherwise).
□
For any set P of size λ<κ and any κ-complete ideal J on P, IEκ2(J) holds.
Suppose that κ is the successor of a singular limit of λ-compact cardinals. Then by a result of Magidor and Shelah [27], κ has the tree property, and in fact, as shown in [10], TP(κ,λ′) holds for every cardinal λ′≥κ. We modify the proof so as to obtain the following which improves a result of [11].
PROPOSITION 7.14**.**
Suppose that κ=ν+, where ν is a singular limit of mildly λ<κ-ineffable cardinals. Then PS(κ,λ) holds.
Proof. Set σ=cf(ν), and select an increasing sequence ⟨νi:i<σ⟩ of mildly λ<κ-ineffable cardinals with σ<ν0 and sup{νi:i<σ}=ν. Suppose that for each a∈Pκ(λ), Qa is a partition of Pκ(λ) with ∣Qa∣≤ν. For a∈Pκ(λ), pick an onto function ψa from Qa to ν, and let Wap denote the set of all c∈Pκ(λ) such that p= the least r such that c∈⋃ψa‘‘r. By Fact 7.13, we may find t:Pκ(λ)→σ and a ν0-complete ideal K on Pκ(λ) extending Iκ,λ such that {Wat(a):a∈Pκ(λ)}⊆K∗. There must be S∈K+ and p<σ such that t takes the constant value p on S. Thus Wap∩Wxp∈Iκ,λ+ for all a,x∈S. Define g:S×S→Pκ(λ) so that a∪x⊆g(a,x) and g(a,x)∈Wap∩Wxp. Further define h:S×S→νp×νp so that g(a,x)∈ψa(α)∩ψx(β), where h(a,x)=(α,β). For a∈S and (α,β)∈νp×νp, put Ta(α,β)={x∈S:h(a,x)=(α,β)}. By Fact 7.13, we may find u:S→νp×νp and a νp+1-complete ideal G on Pκ(λ) extending Iκ,λ∣S such that {Tau(a):a∈S}⊆G∗. There must be A∈G+ and (α,β)∈νp×νp such that u is constantly (α,β) on A. Pick X∈Iκ,λ+∩P(A) so that A∖X∈Iκ,λ+. Set B=X if A∖X∈G+, and B=A∖X otherwise. Now given a,b∈B, pick x∈Ta(α,β)∩Tb(α,β)∩(A∖B) with a∪b⊆x. Then clearly, g(a,x)∈ψa(α)∩ψx(β) and g(b,x)∈ψb(α)∩ψx(β).
□
8 Distributivity
OBSERVATION 8.1**.**
Given a κ-complete, fine ideal J on Pκ(λ), the following are equivalent :
(i)
J* is (λ<κ,2)-distributive.*
2. (ii)
Given A∈J+ and F:A×A→λ<κ with the property that ∣{F(a,b):b∈A}∣<κ for all a∈A , there is B∈J+∩P(A) and h:B→λ<κ such that {b∈B:F(a,b)=h(a)}∈J for all a∈B.
3. (iii)
J+J(J+)ρn* holds whenever 2≤n<ω and 0<ρ<κ.*
4. (iv)
J+J(J+)23* holds.*
Proof. (i) → (ii) : Use Observation 5.4.
(ii) → (iii) : Assume that (ii) holds. It is readily seen that J+J(J+)2 (and hence J+J(J+,κ)2) holds. By Observation 5.18, it follows that κ is weakly compact (and therefore inaccessible). Now to prove (iii), we proceed by induction on n. For n=2, the assertion easily follows from (ii). Suppose now that the assertion has been verified for a certain n. Fix A∈J+ and F:An+1→ρ, where 2≤ρ<κ. For b∈A, let Tb denote the collection of all functions t from (A∩P(b))n−1 to ρ. Notice that ∣Tb∣<κ. Define G:A×A→⋃b∈ATb as follows. Given (b,c)∈A×A, let G(b,c) be the element t of Tb defined by t(a1,⋯,an−1)=F(a1,⋯,an−1,b,c). We may find B∈J+∩P(A), h∈∏b∈BTb, and Xb∈J for b∈B such that G(b,c)=h(b) whenever b,c∈B and c∈/Xb. Define H:⋃b∈B((A∩P(b))n−1×{b})→ρ by H(a1,⋯,an−1,b)=h(b)(a1,⋯,an−1). There must C∈J+∩P(B), i<2, and Ya∈J for a∈C with
{b∈Pκ(λ):a∖b=∅}⊆Ya
such that H(a1,⋯,an−1,an)=i whenever a1,⋯,an∈C and am+1∈/Ya1∪⋯∪Yam for 1≤m<n. For a∈C, put Za=Xa∪Ya. Then clearly, F(a1,⋯,an+1)=G(an,an+1)(a1,⋯,an−1)=h(an)(a1,⋯,an−1)=H(a1,⋯,an−1,an)=i whenever a1,⋯,an+1∈C and am+1∈/Za1∪⋯∪Zam for 1≤m≤n.
(iii) → (iv) : Trivial.
(iv) → (i) : Assume that (iv) holds. Then by Observation 5.18, κ is inaccessible. Now let A∈J+, and Wa⊆Pκ(λ) for a∈Pκ(λ). Define F:A×A×A→2 by : F(a,b,c)=0 if and only if {d⊆a:b∈Wd}={d⊆a:c∈Wd}. There must be B∈J+∩P(A), i<2, and Za∈J for a∈B such that F(a,b,c)=i whenever a,b,c∈B, b∈/Za and c∈/Za∪Zb. Given a∈B, we may find C∈J+∩P(B) such that {d⊆a:b∈Wd}={d⊆a:c∈Wd} whenever d⊆a and b,c∈C. It easily follows that i=0. Now fix d∈Pκ(λ). Pick a∈B with d⊆a, and b∈B∖Za. Then either B∖(Za∪Zb)⊆Wd, or B∖(Za∪Zb)⊆Pκ(λ)∖Wd.
□
Note the similarity with Fact 7.3 (ii) or Fact 7.12. One could indeed argue that (λ<κ,2)-distributivity of J (respectively, mild λ<κ-ineffability of κ) makes more sense (or is more natural) than (λ,2)-distributivity (respectively, mild λ-ineffability). However the question of Abe mentioned in the introduction concerns (λ,2)-distributivity (and not (λ<κ,2)-distributivity) of Iκ,λ, so let us focus on (λ,2)-distributivity.
PROPOSITION 8.2**.**
Suppose that J is a (κ,2)-distributive, κ-normal, fine ideal on Pκ(λ). Then J↾κ is (κ,2)-distributive.
Proof. Let X∈(J↾κ)+, and Wα⊆κ for α<κ. For α<κ, set Tα={a∈Ωκ,λ:a∩κ∈Wα}. We may find h∈∏α<κ{Tα,Pκ(λ)∖Tα}, B∈J+∩P({a∈Ωκ,λ:a∩κ∈X}), and Zα∈J for α<κ such that B∖Zα⊆h(α) for all α<κ. Define k∈∏α<κ{Wα,κ∖Wα} by : k(α)=Wα if and only if h(α)=Tα. Put S={a∈B:∀α∈a∩κ(a∈/Zα)} and Y={a∩κ:a∈S}. It is easy to see that Y∈(J↾κ)+∩P(X). Furthermore Y∖(α+1)⊆k(α) for all α<κ.
□
OBSERVATION 8.3**.**
Suppose that dκ≤λ=cov(λ,κ+,κ+,κ). Then for any D∈NSκ,λ∗ and any X∈NSκ+∩NCIκ, Iκ,λ∣{a∈D:sup(a∩κ)∈X} is not (κ,2)-distributive.
Proof. Given D∈NSκ,λ∗ and X∈NSκ+∩NCIκ, set K=Iκ,λ∣{a∈D:sup(a∩κ)∈X}. Assume toward a contradiction that K is (κ,2)-distributive. By Fact 5.31, there is C∈NSκ,λ∗ such that Iκ,λ∣C is κ-normal. Put J=K∣C. Then clearly, J is κ-normal and (κ,2)-distributive. Hence by Observation 5.21 and Proposition 8.2, J↾κ is a normal, (κ,2)-distributive, fine ideal on κ. It follows that X∈J↾κ. Contradiction.
□
PROPOSITION 8.4**.**
Suppose that dκ≤λ and cf(λ)=κ. Then for any D∈NSκ,λ∗ and any X∈NSκ+∩NCIκ, Iκ,λ∣{a∈D:sup(a∩κ)∈X} is not (λ,2)-distributive.
Proof. By Fact 7.4, Proposition 7.12 and Observation 8.3.
□
9 The case cf(λ)=κ
By Fact 4.4, if cf(λ)=κ, then cov(λ,κ+,κ+,κ)>λ. Thus a different approach is needed in case cf(λ)=κ.
PROPOSITION 9.1**.**
Let J be a κ-complete, fine ideal on Pκ(λ) such that cof(J)<max{dκ,u(κ+,λ)}. Then the following hold :
(i)
Let Aα∈J+ for α<κ be given such that Aα⊆Aβ whenever β<α<κ. Then there is C∈J+ such that C∖Aα∈Iκ,λ for all α<κ.
2. (ii)
Suppose that κ is weakly compact. Then given Wα⊆Pκ(λ) for α<κ, there is C∈J+ and h∈∏α<κ{Wα,Pκ(λ)∖Wα} such that C∖h(α)∈Iκ,λ for all α<κ.
Proof. (i) : The proof is a straightforward modification of that of Proposition 2.7 in [23].
(ii) : Assume that κ is weakly compact, and let Wα⊆Pκ(λ) for α<κ. There must be h∈∏α<κ{Wα,Pκ(λ)∖Wα} such that Aα∈J+ for all α<κ, where Aα=⋂β≤αh(β). By (i) we may find C∈J+ such that C∖Aα∈Iκ,λ for all α<κ. Then clearly, C∖h(α)∈Iκ,λ for every α<κ.
□
COROLLARY 9.2**.**
Suppose that κ is weakly compact, and let J be a κ-complete, fine ideal on Pκ(λ) such that cof(J)<max{dκ,u(κ+,λ)}. Then the following hold :
(i)
J* is (κ,2)-distributive.*
2. (ii)
J+Iκ,λκ(J+)ρ2* holds for any nonzero cardinal ρ<κ.*
Proof. Use Observation 5.4.
□
FACT 9.3**.**
([18], [27])* Suppose that κ is weakly inaccessible, and let J be a κ-complete, fine ideal on Pκ(λ) such that cof(J)<max{dκ,u(κ+,λ)}. Then J+Iκ,λκ[J+]κ+2 holds.*
To obtain a negative partition relation, we will go one cardinal up and work with partitions of κ+×Pκ(λ).
DEFINITION 9.4**.**
Given 2≤n<ω, two collections X and Y of subsets of Pκ(λ), an ideal J on Pκ(λ), and a cardinal ρ, XYκ+(J+)ρ2 means that for any F:κ+×Pκ(λ)→ρ and any A∈X, there is i<ρ and B∈J+∩P(A) such that {b∈B:F(sup(a∩κ+),b)=i}∈Y for all a∈B.
Then {C∩Sκ,λλ}Iκ,λκ+[Iκ,λ+]λ2 fails for some C∈NSκ,λ∗.
PROPOSITION 9.6**.**
Suppose that cf(λ)=κ and 2κ≤λ. Let D∈NSκ,λ∗ and J=Iκ,λ∣(D∩Sκ,λλ). Then the following hold :
(i)
J+J(J+)ω2* does not hold.*
2. (ii)
J* is not (λ,2)-distributive.*
Proof. (i) : By Observations 5.18 and 6.4 and Fact 9.5.
(ii) : Suppose otherwise. Then λ<κ=λ by Fact 7.10, since κ is mildly λ-ineffable by Fact 7.4. Hence J+J(J+)ω2 holds by Observation 8.1, which contradicts (i).
□
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