Martin's maximum and the non-stationary ideal
Shimon Garti Yair Hayut Haim Horowitz Menachem Magidor

<ref>

We deal with NS_ω_1 and 𝒟_ω_1 under Martin's maximum and the proper forcing axiom.


Topological π-junctions from crossed Andreev reflection in the Quantum Hall regime
    F. Finocchiaro^1,2, F. Guinea^2,3 and P. San-Jose^1
    December 30, 2023
==================================================================================







§ INTRODUCTION


In this paper we try to study the influence of forcing axioms on the non-stationarty ideal over ℵ_1 (denoted by NS_ω_1) and its dual, the club filter 𝒟_ω_1.
In the first section we introduce a general reflection theorem from which we derive a saturation property of NS_ω_1.
In the second section we deal with Galvin's property with respect to 𝒟_ω_1.


 Saturation. An ideal 𝒥 over κ is (μ,λ,θ)-saturated iff for every collection {A_α:α<μ}⊆𝒥^+ there is a sub-collection {A_α_ε:ε∈λ} such that ⋂{A_α_ε:ε∈ y}∈𝒥^+ for every y∈[λ]^θ.


An ideal ℐ over a successor cardinal κ^+ is Laver iff ℐ is uniform (i.e., contains all the bounded subsets of κ^+), κ^+-complete and (κ^++,κ^++,κ)-saturated. If we require only (κ^++,κ^+,κ)-saturation then ℐ is called weakly Laver. If we have just (κ^++,κ^+,<κ)-saturation then ℐ is almost weakly Laver.

Laver ideals can be forced over ℵ_1 by collapsing sufficiently large cardinals. In the original work of Laver, <cit.>, a huge cardinal was used. Other concepts of saturation can be forced with the same basic idea, i.e. to collapse a large cardinal and preserve some of its qualities.
An interesting question is what happens under classical forcing axioms. One expects ℵ_1 to behave like a large cardinal under these axioms.

The simplest notion of saturation for an ideal ℐ over κ^+ says that if {A_α:α<κ^++}⊆ℐ^+ then there are α<β<κ^++ so that A_α∩ A_β∈ℐ^+. Such an ideal is called Kunen, and the following theorem from <cit.> embodies the above expectation:


 Under MM, the ideal NS_ω_1 is Kunen.


thmkunen

So strong forcing axioms yield saturation properties at ℵ_1. However, the situation is more involved. Laverness, for example, is excluded from ℵ_1 under any classical forcing axiom. The theorem below is typical, its first assertion is due to Larson, <cit.>, and its second part appears in <cit.>.


 Let 𝒥 be a uniform ideal.


  (ℵ) If MA + 2^ω=ω_2 and 𝒥 is an ideal over ℵ_1 then 𝒥 is not Laver.

  (ℶ) If BA + 2^ω_1=ω_3 and 𝒥 is an ideal over ℵ_2 then 𝒥 is not Laver.



thmmaba

A natural question, raised by Larson, is about the intermediate concept of weak Laverness at ℵ_1 under some forcing axiom. Larson asked, in particular, about NS_ω_1.
In the first section we obtain the following result:


 If MM holds then NS_ω_1 is almost weakly Laver.


thmawlaver

We indicate that this result follows in a different way from <cit.>, Lemma 3.10.
In the second section we address the cardinal characteristic 𝔤𝔭, related to 𝒟_ω_1.
We also deal with Galvin's property under the proper forcing axiom.
Regarding this property we prove the consistency of PFA with the strong negation of Galvin's property.
We believe that this strong negation holds in any model of PFA.

We shall use the standard notation. If ℐ is an ideal over κ^+ then ℐ will be uniform and κ^+-complete unless otherwise stated. The set 𝒫(κ^+)∖ℐ is denoted by ℐ^+ (the ℐ-positive sets). We employ the Jerusalem forcing notation, i.e. p≤ q reads p is weaker than q.

If g:κ→ S and β∈κ then g”β = {g(α):α<β}.
The set 𝒫_κλ is the collection of subsets of λ whose cardinality is less than κ.
Let M,N be models of (enough) ZFC. We say that N is a κ-end-extension of M iff for every ζ∈ N∩κ either ζ∈ M or ζ≥sup(M∩κ). This concept will be used below with respect to κ=ℵ_2.

Let A be a set and let F:[A]^<ω→𝒫_κ(A) be a function. An element x of 𝒫_κ(A) will be called a closure point of F iff F(e)⊆ x for every e∈[x]^<ω. The set of closure points of F will be denoted by cℓ_F. It is straightforward that this set is a club in 𝒫_κ(A). Moreover, every club C of 𝒫_κ(A) can be coded by a function F in the sense that there exists a function F:[A]^<ω→𝒫_κ(A) such that cℓ_F⊆ C.

A central concept from <cit.> is ℵ_1-reflection. Suppose that S⊆[ℋ(θ)]^ω. We shall say that S reflects to a set of size ℵ_1 iff there exists x⊆ℋ(θ), |x|=ℵ_1, ω_1⊆ x such that S∩[x]^ω is stationary in [x]^ω. The following is labeled in <cit.> as Theorem 13:


 MM and ℵ_1-reflection. Assume MM and let θ be a regular uncountable cardinal. Every stationary set S⊆ [ℋ(θ)]^ω reflects to a set of size ℵ_1.


thmreflection





§ ALMOST WEAK LAVERNESS


Our first statement is a strengthening of typical properties which can be proved under MM. It says, roughly, that we have a good control on M∩ω_1 for countable elementary submodels of ℋ(θ). We shall need, however, a simple lemma ahead of the main claim.


 Let ℚ be an ℵ_2-cc forcing notion, and let G⊆ℚ be generic.
Let {q_ε:ε∈ω_2} be any collection of conditions in ℚ. There exists a condition p∈ℚ which forces ℵ_2-many members from this collection into the generic set G.


Proof. Fix a collection {q_ε:ε∈ω_2}⊆ℚ.
We shall prove that some q_ε forces that |{ζ:q_ζ∈G}|=ℵ_2.
Toward contradiction assume that there is no such condition.
Hence for every ε∈ω_2 there are p_ε∈ℚ and an ordinal α_ε such that q_ε≤ p_ε and p_ε⊩α_ε is the first ordinal such that q_ζ∉G for every ζ∈[α_ε,ω_2). Let δ = ⋃_ε∈ω_2α_ε.

Case 1: δ=ω_2.

In this case there is an increasing sequence ⟨ε_ζ: ζ∈ω_2⟩ of ordinals in ω_2 such that ⟨α_ε_ζ:ζ∈ω_2⟩ is strictly increasing. We claim that the corresponding set {p_α_ε_ζ : ζ∈ω_2} is an antichain in ℚ, contradicting the ℵ_2-chain condition.
For proving this fact assume that η<ζ<ω_2. So p_α_ε_η⊩ q_ζ∉G for every ζ∈[α_ε_η,ω_2) while p_α_ε_ζ⊩∃τ∈ [α_ε_η,ω_2), q_τ∈G. It means that p_α_ε_η⊥ p_α_ε_ζ, so we are done.

Case 2: δ<ω_2.

In this case there is an ordinal α<ω_2 such that |{ε∈ω_2: α_ε=α}|=ℵ_2. Choose any ζ∈ω_2 such that α<ζ and α_ζ=α.
On the one hand p_ζ⊩ q_ζ∈G since q_ζ≤ p_ζ. On the other hand, p_ζ forces that q_τ∉G whenever τ∈[α_ζ,ω_2)=[α,ω_2). In particular, p_ζ⊩ q_ζ∉G, a contradiction.

lemchain

The main claim, to be proved below, says that one can find nice extensions of countable elementary submodels. We shall prove it in two steps, first relativized to some club subset and then in a general form. Ahead of the claims (and proofs) we define the properties that we shall need.


 Nice extensions. Assume that M is countable and M≺ℋ(θ) for some regular cardinal θ.
Assume further that {A_β:β∈ω_2}⊆ NS^+_ω_1.
An elementary submodel N≺ℋ(θ) will be called a nice extension of M iff:


  (ℵ) M⊆ N.

  (ℶ) N is countable.

  (ℷ) N is an ω_2-end-extension of M.

  (ℸ) ∃β∈ N∖ M such that M∩ω_1 = N∩ω_1∈ A_β.



The third requirement, namely N is an ω_2-end-extension of M, can be verified while concentrating on ordinals of ω_1. This is the content of the following:


 Suppose that θ=(θ) and M,N are countable structures which satisfy M≺ N≺ℋ(θ). If M∩ω_1=N∩ω_1 then N is an ω_2-end-extension of M.


Proof. Assume that ξ∈ N∩ω_2 and ζ∈ M∩ω_2 for some ζ>ξ. We need showing that ξ∈ M. If ξ∈ω_1 then ξ∈ M as we are assuming that M∩ω_1 = N∩ω_1. So we may assume that ω_1≤ξ, and hence ζ>ω_1.

Now ζ∈ω_2∩ M and ζ>ω_1 so there is a bijection h:ζ→ω_1, h∈ M. Since ξ<ζ we see that h(ξ)∈ω_1. Since ξ∈ N and h∈ M⊆ N we see that h(ξ)∈ N. Therefore, h(ξ)∈ N∩ω_1 = M∩ω_1, hence h(ξ)∈ M as well. By elementarity h^-1∈ M and hence ξ=h^-1(h(ξ))∈ M, so we are done.

lemextension

We are ready now for the first assertion:


 Assume MM. Suppose that {A_γ: γ∈ω_2}⊆ NS^+_ω_1, A∈ NS^+_ω_1 and every stationary subset of A intersects ω_2-many A_γ-s positively. Then for every sufficiently large regular cardinal θ there is a club C⊆𝒫_ω_1(ℋ(θ)) which consists of countable elementary submodels of ℋ(θ) such that for every M∈ C if M∩ω_1∈ A then there exists a nice extension N of M.


Proof. Toward contradiction assume that such a club C does not exist. Fix a stationary subset S of 𝒫_ω_1(ℋ(θ)) such that every M∈ S is a countable elementary submodel of ℋ(θ) satisfying M∩ω_1∈ A and there is no nice extension of M.

By the reflection principle of Theorem <ref> there is some I⊆ℋ(θ), |I|=ℵ_1, ω_1⊆ I such that S∩𝒫_ω_1(I) is stationary in 𝒫_ω_1(I).
Fix a bijection g:ω_1→ I. Define T = {β∈ω_1: g”β∈ S}. We claim that T is a stationary subset of ω_1, and moreover T∩ A is stationary. For suppose not, and choose a club D⊆ω_1 such that D∩(T∩ A)=∅. Without loss of generality if α∈ D then ω_1∩ g”α=α, as we can cut D with the club of ordinals α for which ω_1∩ g”α=α.

Let E = {M∈𝒫_ω_1(I): M∩ω_1∈ D, g”(M∩ω_1)⊆ M ∧ g^-1[M]⊆ M}. Notice that E is a club in 𝒫_ω_1(I). Likewise, M=g”(M∩ω_1) whenever M∈ E. For this, if x∈ M then x=g(g^-1(x)). But g^-1(x)∈ M∩ω_1 since M is closed under g^-1, so g(g^-1(x))∈ g”(M∩ω_1). It means that M⊆ g”(M∩ω_1), and g”(M∩ω_1)⊆ M follows from the fact that M is closed under g.

Pick any element M∈ E∩ S. On the one hand M∩ω_1∈ D since M∈ E. On the other hand M∩ω_1∈ T∩ A since M∈ S and M=g”(M∩ω_1). In other words, the ordinal M∩ω_1 belongs to D∩(T∩ A), a contradiction.

Define η=sup(I∩ω_2). Let D' = {β∈ω_1: ω_1∩ g”β = β}, and let B' = (T∩ A)∩ D'. The set B'⊆ A is stationary, and by the properties of A there is an ordinal γ∈ω_2 such that η<γ and B = B'∩ A_γ is stationary in ω_1. Choose N≺ℋ(θ), |N|=ℵ_0 such that g,γ∈ N and β=N∩ω_1∈ B. Set M_0=g”β. Notice that M_0∈ S since β∈ B⊆ T. We claim that N is a nice extension of M_0, which is an absurd as M_0∈ S.

First observe that M_0⊆ N since g∈ N and β⊆ N. Second, N∩ω_1=β=M_0∩ω_1 since β∈ D', so Lemma <ref> ensures that N is an ω_2-end-extension of M_0. Finally, γ∈ N by the choice of N but γ∉ M_0. Indeed, M_0=g”β⊆ I so if γ∈ M_0 then γ=g(α) for some α<β. It means that γ∈ I (as g:ω_1→ I) and γ∈ω_2, which is impossible since γ>η=sup(I∩ω_2).
Summing up, N is a nice extension of the element M_0∈ S, a contradiction.

clmclub

We proceed to the second step, which is basically the same statement but without applying to the club C⊆𝒫_ω_1(ℋ(θ)).


 Nice extensions. Assume MM and Suppose that {A_γ: γ∈ω_2}⊆ NS^+_ω_1, A∈ NS^+_ω_1, and for every S∈ NS^+_ω_1, S⊆ A there are ω_2-many ordinals β for which S∩ A_β∈ NS^+_ω_1. Then for every large enough regular cardinal χ and every countable M≺ℋ(χ) so that M∩ω_1∈ A one can find a nice extension N≺ℋ(χ).


Proof. We shall assume that ℋ(χ) is augmented by a fixed well-ordering and any required functions or predicates, e.g. definable Skolem functions and the sequence {A_γ: γ∈ω_2} as well as the set A as predicates. Let θ=(θ) be large enough to satisfy Claim <ref>. We may assume that every function from the finite subsets of ω_2 into ω_2 belongs to ℋ(θ), upon noticing that this requires θ≥ω_3. Choose χ=(χ)>θ so that ℋ(θ)∈ℋ(χ). The latter is needed in order to guarantee that the club C from Claim <ref> belongs to ℋ(χ).

Assume that M≺ℋ(χ), |M|=ℵ_0 and M∩ω_1∈ A. It follows that M∩ℋ(θ)∈𝒫_ω_1(ℋ(θ)), and we may assume that M∩ℋ(θ)∈ C. Indeed, we can assume that the club C is coded by a function F which appears as a predicate in the augmented structure of ℋ(χ). Now F is a function from ℋ(θ)^<ω into ℋ(θ) and M is an elementary submodel of ℋ(χ) with this predicate, so together M∩ℋ(θ) is closed under F and hence belongs to C.

By virtue of Claim <ref> there is a nice extension N_0 of M∩ℋ(θ), and in particular N_0≺ℋ(θ). We wish to extend N_0 to an elementary submodel of ℋ(χ) which will be a nice extension of M.

Define N= Sk^ℋ(χ)(M∪(N_0∩ω_2)). Notice that N≺ℋ(χ), and we shall prove that N is as desired. Being the Skolem hull of a countable set, N is virtually countable. Clearly, M⊆ N. Likewise, there exists an ordinal β∈ N_0∖(M∩ℋ(θ)) such that (M∩ℋ(θ))∩ω_1 = N_0∩ω_1∈ A_β. Since θ is large enough, (M∩ℋ(θ))∩ω_1 = M∩ω_1, and β∈ N_0∩ω_2⊆ N. So if we prove that N_0∩ω_2 = N∩ω_2 we will conclude that N_0∩ω_1 = N∩ω_1∈ A_β. Moreover, N_0∩ω_2 = N∩ω_2 accomplishes all the requirements listed in the definition of a nice extension.

Let us prove, therefore, that N_0∩ω_2 = N∩ω_2. Clearly, N_0∩ω_2 ⊆ N∩ω_2, so assume that ξ∈ N∩ω_2 with the goal of showing that ξ∈ N_0 as well.
By the definition of N there exists a Skolem function f, a finite set {a_1,…,a_k}⊆ M and a finite set of ordinals {γ_1,…,γ_ℓ}⊆ N_0∩ω_2 such that:

    ξ = f(a_1,…,a_k,γ_1,…,γ_ℓ).

We may assume that Rang(f)⊆ω_2 (if not, replace f by g which agrees with f whenever f(e)∈ω_2 and assumes zero otherwise). Define h:[ω_2]^ℓ→ω_2 by h(t) = f(a_1,…,a_k,t).

We claim that h∈ M.
Indeed, M≺ℋ(χ) and ℋ(χ) contains a complete set of Skolem functions, including the specific function f. The function h is definable from f and {a_1…,a_k}⊆ M hence belongs to M.
Likewise, assuming that θ is sufficiently large we may assume that all the functions from [ω_2]^ℓ into ω_2 are in ℋ(θ), in particular h∈ℋ(θ) and hence h∈ M∩ℋ(θ)⊆ N_0. Since {γ_1,…,γ_ℓ}⊆ N_0 we conclude that ξ = f(a_1,…,a_k,γ_1,…,γ_ℓ) = h(γ_1,…,γ_ℓ)∈ N_0, and the proof is accomplished.

mclm


 The ability to produce nice extensions of countable elementary submodels resembles similar theorems which hold at measurable cardinals. It means that ℵ_2 behaves like a measurable cardinal under MM. This is coherent with other statements which show that small accessible cardinals possess large cardinal properties under strong forcing axioms.


rmeasurable

As indicated in the introduction, the fact that NS_ω_1 is almost weakly Laver under MM follows from Lemma 3.10 of <cit.>.
It can also be derived from the main claim above:


 The ideal NS_ω_1 is (ℵ_2,ℵ_1,<ℵ_0)-saturated under MM.


Proof. Suppose that {A_α:α∈ω_2}⊆ NS^+_ω_1. Let A⊆ω_1 be as guaranteed by Lemma <ref>. By induction on γ∈ω_1 we choose (for sufficiently large χ) a model N_γ≺ℋ(χ) as follows:

γ=0: Let N_0 be a countable elementary submodel of ℋ(χ) so that N_0∩ω_1∈ A. We also assume that {A_α:α∈ω_2}∈ N_0.

γ+1: Denote N_γ by M, and choose N as described in Claim <ref> with respect to M. Let α_γ∈ N∖ M be the ordinal given by the definition of nice extensions. Set N_γ+1 = N.

γ is limit: Let N_γ = ⋃_δ<γN_δ.

Denote the union ⋃_γ∈ω_1N_γ by N. Notice that {A_α_γ:γ∈ω_1}⊆ N. We claim that this collection exemplifies the almost weak Laverness of NS_ω_1. For this, let u be a finite subset of ω_1 and S = ⋂_γ∈ uA_α_γ. Notice that S∈ N as u is finite.

Assume toward contradiction that S∈ NS_ω_1, and let C be a club subset of ω_1 so that C∈ N and N C∩ S=∅. Let τ be the characteristic ordinal N_0∩ω_1. By the nature of the N_γ-s we see that N_γ∩ω_1=τ for every γ∈ω_1 and hence N∩ω_1=τ.

On the one hand, τ∈ C since ℋ(χ) believes that C is unbounded below τ and C is closed. On the other hand, τ∈ A_α_γ for each γ∈ u since N_γ∩ω_1 = N∩ω_1 = τ for every γ∈ω_1 and by the definition of nice extensions. In particular τ∈ S, a contradiction.

thmmt

We make the comment that the finite number of A_α_β-s is required only in order to make sure that S = ⋂_β∈ uA_α_β∈ N. This seems to be an essential obstacle while considering the possibility of strengthening the above theorem. Indeed, no closure assumption on N can be made (apart from the closure to finite sequences which follows from elementarity).

The question of weak Laverness under MM remains open, but almost weak Laverness is quite settled. It is interesting to compare MM with PFA at this point:


 Forcing axioms and the non-stationary ideal.


  (ℵ) NS_ω_1 is almost weakly Laver under MM.

  (ℶ) The almost weak Laverness of NS_ω_1 is independent over the PFA.



Proof. The first statement is the above theorem, and PFA is consistent with NS_ω_1 being almost weakly Laver since MM implies PFA. The consistency of PFA with the failure of almost weak Laverness of NS_ω_1 and even with the failure of the simplest saturation property is due to Velickovic, <cit.> (and as noted there, this result was obtained independently by Shelah).

corawl


 For the main result we used MM, but actually almost weak Laverness follows from the ℵ_1-reflection principle and the fact that NS_ω_1 is Kunen. This might be meaningful when other axioms are deemed with respect to various saturation properties.






§ THE CLUB FILTER


A celebrated theorem of Galvin which appeared in <cit.> says that under the continuum hypothesis any collection of ℵ_2 many clubs of ℵ_1 contains a sub-collection of size ℵ_1 whose intersection is a club subset of ℵ_1.
Remark that the collection of end-segments of ℵ_1 is a family of ℵ_1 many clubs of ℵ_1, each sub-family of which of size ℵ_1 has an empty intersection.
This means that one can gather a small number of clubs of ℵ_1 with empty intersection of each ℵ_1 of them, but (under the continuum hypothesis) one cannot collect many clubs in this way. Here, many clubs means ℵ_2 clubs of ℵ_1.

Galvin's result can be generalized in several directions, and one of them is based on replacing the continuum hypothesis by Devlin-Shelah's weak diamond.
It has been proved in <cit.> that if 2^ω<2^ω_1 (this is equivalent to the weak diamond at ℵ_1) then any collection of λ clubs of ℵ_1 contains a sub-collection of size ℵ_1 whose intersection is a club subset of ℵ_1 where λ=(2^ω)^+.
This is quite a surprising result, since 2^ω_1 can be much larger than λ and yet Galvin's property holds here with respect to any collection of λ sets. Actually, this gives rise to defining the first cardinal which satisfies this property as a cardinal characteristic of the continuum:


 The cardinal characteristic 𝔤𝔭. We define 𝔤𝔭 as the minimal κ such that every family {C_α:α∈κ^+} of club subsets of ℵ_1 contains a subfamily {C_α_β:β∈ω_1} whose intersection is closed and unbounded in ℵ_1.


The definition comes from <cit.> and it has been shown there that 𝔤𝔭 is well-defined and assumes a value between ℵ_1 and 2^ℵ_0.
The behavior of 𝔤𝔭 when related to other cardinal characteristics is very peculiar.
It is consistent that 𝔤𝔭<𝔪, but it is also consistent that 𝔤𝔭>𝔦.
Dealing with 𝔤𝔭 by itself, it has been asked in <cit.> what are the possible cofinalities of 𝔤𝔭 and, in particular, is it consistent that (𝔤𝔭)=ω.
In this paper we try to deal with this question.
Like many other cardinal characteristics, the definition of 𝔤𝔭 generalizes to higher cardinals.
This is central in our context, since our knowledge on (𝔤𝔭_κ) for κ=(κ)≥ℵ_3 is a bit better.


 The cardinal characteristic 𝔤𝔭_κ. Let κ=(κ). We define 𝔤𝔭_κ as the minimal λ such that every family {C_α:α∈λ^+} of club subsets of κ^+ contains a subfamily {C_α_β:β∈κ^+} whose intersection is closed and unbounded in κ^+.


A straightforward modification of the statements in <cit.> shows that 𝔤𝔭_κ is well-defined and that κ^+≤𝔤𝔭_κ≤ 2^κ.
For basic background regarding pcf theory we suggest <cit.>, and for further information we refer to <cit.>.

We commence with showing that under some pcf assumptions one can prove that (𝔤𝔭)>ω. The assumption is a consequence of the existence of a good scale, which in turn can be proved to exist from instances of weak square. However, this assumption cannot be established in ZFC, and actually it fails if one assumes certain instances of Chang's conjecture.
We shall prove our first theorem based on a pcf assumption, and later we shall connect this assumption to known combinatorial concepts like weak squares and Chang's conjecture.



Let μ be a singular cardinal.


  (A) Suppose that:


  (a) ω=(μ)<μ<2^ω and (μ_n:n∈ω) is an increasing sequence of regular cardinals such that μ=⋃_n∈ωμ_n.

  (b) J⊇ J^ bd_ω and tcf(∏_n∈ωμ_n,J)=μ^+, as witenessed by the scale (f_α:α∈μ^+).

  (c) For every A⊆μ^+ such that |A|=ℵ_1 it is true that |{f_α(n):α∈ A,n∈ω}|=ℵ_1.

Then 𝔤𝔭≠μ.

  (B) Suppose that:


  (a) ω_1=(μ)<μ≤ 2^ω and (μ_γ:γ∈ω_1) is an increasing sequence of regular cardinals such that μ=⋃_γ∈ω_1μ_γ.

  (b) J⊇ J^ bd_ω_1 and tcf(∏_γ∈ω_1μ_γ,J)=μ^+, as witenessed by the scale (g_α:α∈μ^+).

  (c) For every A⊆μ^+ such that |A|=ℵ_1 there is δ∈ω_1 such that |{g_α(γ):α∈ A,γ∈δ}|=ℵ_1.

Then 𝔤𝔭≠μ.



Proof. For the first part, assume toward contradiction that 𝔤𝔭=μ.
Fix a collection 𝒟={D_α:α∈μ} such that ⋂{D_α_β:β∈ B} is bounded in ω_1 for every B∈[μ]^ℵ_1.
For every α∈μ^+ let C_α=⋂{D_f_α(n):n∈ω}.
Each C_α is a club subset of ℵ_1, being the intersection of but ℵ_0-many clubs of ℵ_1.
Set 𝒞={C_α:α∈μ^+}.

Since 𝔤𝔭=μ, one can choose A⊆μ^+,|A|=ℵ_1 such that C=⋂{C_α:α∈ A} is a club subset of ℵ_1.
By our assumptions, |{f_α(n):α∈ A, n∈ω}|=ℵ_1 and hence {D_f_α(n):α∈ A,n∈ω}⊆ [𝒟]^ℵ_1.
By the nature of the collection 𝒟, the set D=⋂{D_f_α(n):α∈ A, n∈ω} is bounded in ω_1.
However, D=C, a contradiction.

For the second part, assume toward contradiction that 𝔤𝔭=μ and fix a family of clubs 𝒟={D_α:α∈μ} which exemplifies this fact.
For every α∈μ^+ we define C_α = Δ{D_g_α(γ):γ∈ω_1}, so every C_α is a club subset of ℵ_1.
Let 𝒞={C_α:α∈μ^+}.
Choose a set A⊆μ^+,|A|=ℵ_1 such that ⋂{C_α:α∈ A} is a club of ℵ_1.
Fix an ordinal δ∈ω_1 such that |{g_α(γ):α∈ A, γ<δ}|=ℵ_1.
Define C=⋂{C_α:α∈ A} - (δ+1), so C is still a club of ℵ_1.
Let D=⋂{D_g_α(γ):α∈ A,γ<δ}, so D is bounded in ω_1.
But if ζ∈ C then for every α∈ A and every γ<δ we see that ζ∈ D_g_α(γ), since ζ>δ>γ.
This means that C⊆ D, a contradiction.

thmcofgp

For placing the above theorem in a familiar combinatorial land we must analyze the conditions under which the assumption on A can be made.
Let (f_α:α<δ) be a scale with respect to J=J^ bd_ω (this is our typical case).
We shall say that γ is a good point if there is an unbounded u⊆γ and n∈ω such that ⟨ f_α(m):α∈ u⟩ is strictly increasing for every m≥ n.
A scale (f_α:α<δ) is good iff every γ∈δ of uncountable cofinality is a good point.


 Assume that ω=(μ)<μ<2^ω, and there exists a good scale (f_α:α∈μ^+) with respect to some product of regular cardinals and J=J^ bd_ω. Then 𝔤𝔭≠μ.


Proof. For every β∈μ^+ such that (β)>ω let u_β be an unbounded subset of β as guaranteed by the assumption that (f_α:α∈μ^+) is a good scale.
Assume that A⊆μ^+,|A|=ℵ_1 and β=sup(A).
Without loss of generality, (β)=ω_1.
By shrinking A and u_β if needed we may assume that the elements of A and u_β are cofinally interleaved.

For every α∈ A choose δ^α_0,δ^α_1∈ u_β such that δ^α_0<α and δ^α_1 is the first element of u_β such that α<δ^α_1.
Choose n=n(α)∈ω such that f_δ^α_0(m)<f_α(m) <f_δ^α_1(m) for every m∈[n,ω).
By shrinking A once more we may assume that for every α∈ A we have n(α)=n_0 for some fixed n_0∈ω.

Since (f_α:α∈μ^+) is a good scale, and (β)>ω, there exists n_1∈ω such that ⟨ f_δ(m):δ∈ u_β⟩ is strictly increasing for every m∈[n_1,ω).
Set n=max{n_0,n_1}.
It follows that |{f_α(ℓ):α∈ A, ℓ∈[n,ω)}|=ℵ_1, as required.

clmvgs

The above claim enables us to derive conclusions about the cofinality of 𝔤𝔭 from a weak form of the square principle.
It is known that □^*_μ implies the existence of a good scale (f_α:α∈μ^+).
Actually, it gives a bit more. It is shown in <cit.> that a better scale can be obtained from the weak square.


 Assume □^*_μ for every μ>(μ)=ω below 2^ω. Then (𝔤𝔭)>ω.


corweaks

The above conclusions are stated with respect to the possibility that (𝔤𝔭)=ω, but the real point here seems to be different.
If μ>(μ)=ω and pp(μ) is large then mild assumptions rule out the possibility of 𝔤𝔭=ν for many cardinals ν∈[μ, pp(μ)).
The following is a typical statement:


 Assume that:


  (a) ω=(μ)<μ<2^ω.

  (b) (μ_n:n∈ω) is a sequence of regular cardinals, μ=⋃_n∈ωμ_n and J⊇ J^ bd_ω.

  (c) λ= tcf(∏_n∈ωμ_n,J) and (f_α:α∈λ) is a scale which witnesses this fact.

  (d) μ≤ν<λ.

  (e) For every A∈[λ]^ℵ_1 it is true that |{f_α(n):α∈ A, n∈ω}|=ℵ_1.

Then 𝔤𝔭≠ν. 

Proof. Follow the proof of Theorem <ref>, upon replacing μ by ν and μ^+ by λ.

thmpp

It follows that the consistency of (𝔤𝔭)=ω requires strong assumptions.
We turn to 𝔤𝔭_κ when κ=(κ)>ℵ_0.
If κ>ℵ_0 then theoretically it may occur that (𝔤𝔭_κ)<κ.
Weak assumptions about good scales imply (𝔤𝔭_κ)>κ by a straightforward generalization of the previous statements.
The purpose of the following theorem is to show, in ZFC, that (𝔤𝔭_κ) cannot drop down dramatically without any special assumption.


 Martin's maximum and the cofinality of 𝔤𝔭_κ. Assume that κ=θ^+3. Then (𝔤𝔭_κ)>θ, and moreover if κ<μ and (μ)≤θ then 𝔤𝔭_κ∉[μ, pp(μ)). Similarly, Martin's maximum implies that (𝔤𝔭_θ^+)>θ for every infinite cardinal θ.


Proof. Fix any μ∈(κ,2^κ) such that (μ)≤θ.
For notational simplicity we assume that (μ)=θ, the proof of the case in which (μ)<θ is identical.
Let λ∈[μ, pp(μ)) be any cardinal, and assume toward contradiction that 𝔤𝔭_κ=λ.
Notice that λ^+≤ pp(μ) and hence there is a set 𝔞 = {λ_i:i∈θ}⊆ Reg∩μ such that sup(𝔞)=μ and tcf(∏𝔞,J)=λ^+ for some J⊇ J^ bd_κ.

Choose an increasing cofinal sequence f̅ = (f_α:α∈λ^+) which exemplifies this fact and satisfies S^ gd_θ(f̅) =_𝒟_λ S^λ_θ.
Such a sequence of functions exists by results of Shelah, see e.g. <cit.>, footnote 5.
By shrinking f̅ if needed we may assume that if A⊆λ^+, |A|=κ^+ then |{f_α(i):α∈ A, i∈θ}|=κ^+.
Now the argument of the proof of Theorem <ref> leads to a contradiction.
The additional part under Martin's maximum follows from <cit.>.

thmgpkappa

Chang's conjecture (μ^+,μ)↠(ℵ_1,ℵ_0) implies the existence of sets A⊆μ^+, |A|=ℵ_1 such that |{f_α(n): α∈ A,n∈ω}|=ℵ_0 for some scale (f_α:α∈μ^+) in the relevant context.
But this fact by itself doesn't imply immediately (𝔤𝔭)=ω.
The point is that even under strong instances of Chang's conjecture we might be able to find a set A of size ℵ_1 for which |{f_α(n): α∈ A,n∈ω}|=ℵ_1.
If we can make sure that such A gives rise to a collection of ℵ_1-many clubs with unbounded intersection in ω_1 then (𝔤𝔭)>ω.
One way to guarantee the above property is captured in the following definition:


 Strong Galvin's property. A collection {C_α:α∈λ} of clubs of κ=(κ)>ℵ_0 exemplifies the strong Galvin's property iff there exists a sequence ⟨ H_ε:ε<κ⟩ so that H_ε∈[λ]^λ for every ε<κ, and for every h∈∏_ε<κH_ε the set ⋂{C_h(ε):ε<κ} is a club subset of κ.


The proof of Galvin from <cit.> as well as the proof under the weak diamond in <cit.> give the strong property for λ=(2^ω)^+ and every family of λ clubs in ω_1.
Let us prove that this strong property implies (𝔤𝔭)>ω.


 Assume that 𝔤𝔭_κ=μ. If every family {C_α:α∈μ^+} of clubs of κ^+ has the strong Galvin's property then (μ)>κ.


Proof. Assume toward contradiction that θ=(μ)≤κ.
Choose a collection 𝒟={D_α:α<μ} which exemplifies 𝔤𝔭_κ=μ.
Fix a scale (f_α:α∈μ^+) in (∏_η∈θμ_η,J^ bd_θ).
By chopping an initial segment and thinning out the sequence we can assume that κ<μ_0 and ζ<η⇒μ_ζ<μ_η.
Denote the ideal J^ bd_θ by J.

For every α∈μ^+ let C_α = ⋂{D_f_α(η):η∈θ}, and let 𝒞 = {C_α:α∈μ^+}.
By our assumptions there is a sequence of sets ⟨ H_ε:ε<κ^+⟩, each H_ε belongs to [μ^+]^μ^+ and for every h∈∏_ε<κ^+H_ε the set ⋂{C_h(ε):ε<κ^+} is a club of κ^+.

We define a set A⊆μ^+,|A|=κ^+ as follows.
By induction on ε<κ^+ we choose an ordinal α_ε∈ H_ε such that sup{f_α_ζ: ζ<ε}<_J f_α_ε.
Since κ<μ_η for every η, the function sup{f_α_ζ: ζ<ε} is an element of the product.
The existence of f_α_ε follows from the fact that |H_ε|=μ^+ and hence {f_α:α∈ H_ε} is cofinal in the product (∏_η∈θμ_η,J).

Let A={α_ε:ε∈κ^+}.
By the choice of each α_ε we see that |{f_α(η): α∈ A, η∈θ}|=κ^+.
As before, let C=⋂{C_α:α∈ A}, D=⋂{D_f_α(η): α∈ A, η∈θ}, so C=D.
However, D is bounded in κ^+ while C is a club of κ^+, a contradiction.

clmsgp

The above claim invites an additional parameter within the definition of 𝔤𝔭.
For κ=(κ)>ℵ_0 and μ≥κ we shall say that a family {C_α:α∈λ} of clubs of κ has μ-Galvin's property iff there is a sequence ⟨ H_ε:ε<κ⟩, H_ε∈[λ]^μ for every ε<κ, such that for every h∈∏_ε<κH_ε the set ⋂{C_h(ε): ε<κ} is a club subset of κ.


 𝔤𝔭_κ with a parameter. Let κ be a regular cardinal, μ∈[κ^+,(2^κ)^+]. The characteristic 𝔤𝔭_κ with parameter μ is the minimal cardinal λ such that every collection {C_α:α∈λ^+} of clubs of κ^+ has μ-Galvin's property. If μ=(2^κ)^+ then we call it the strong 𝔤𝔭_κ and denote it by 𝔰𝔤𝔭_κ.


In this light, Claim <ref> says that the cofinality of 𝔰𝔤𝔭_κ is always larger than κ.
Therefore, if we can force (𝔤𝔭_κ)=κ then we can distinguish 𝔰𝔤𝔭_κ from 𝔤𝔭_κ.
This can be phrased in a general manner:


 Is it consistent that 𝔤𝔭 is strictly smaller than 𝔰𝔤𝔭? More generally, for κ=(κ) and κ^+≤μ_0<μ_1≤(2^κ)^+, is it consistent that 𝔤𝔭_κ with parameter μ_0 is strictly smaller than 𝔤𝔭_κ with parameter μ_1?


In closing this paper we address another problem from <cit.>, about the connection between the proper forcing axiom and Galvin's property. Here we refer to the original property proved by Galvin under the continuum hypothesis, namely that every family of ℵ_2 many clubs of ℵ_1 admits a sub-family of size ℵ_1 whose intersection is a club of ℵ_1.
We shall prove that consistently this property fails under PFA.
For this end, we shall use the usual way to force PFA from a supercompact cardinal.
It is plausible that the negation of Galvin's property follows from the PFA no matter how it is forced.


 PFA and Galvin's property. It is consistent that PFA holds and there is a family {B_α:α∈ω_2} of clubs of ω_1 for which any subfamily of size ℵ_1 has finite intersection.


Proof. We denote the ground model ahead of forcing the proper forcing axiom by V.
Let κ be supercompact in V, and force PFA in the usual way described first by Baumgartner.
In the generic extension, κ becomes ℵ_2 and the PFA holds.
Let 𝔹 be Baumgartner's forcing from <cit.> to add a club of ℵ_1 with finite conditions as defined in V.
Let f be the Laver function employed in the iteration which forces PFA.
Denote the iteration which forces PFA by ⟨ℙ_α,ℚ_β:α≤κ,β<κ⟩, and fix a generic subset G⊆ℙ_κ.

Let A={ζ∈κ: ⊩_ℙ_ζ f(ζ)=𝔹̌}.
If ζ∈ A then ℚ_ζ adds a club of ℵ_1, denoted by 𝔹_ζ.
We claim that the sequence ⟨𝔹_ζ:ζ∈ A⟩ exemplifies the negation of Galvin's property in the generic extension by ℙ_κ.
Remark that in the generic extension this is a set of size ℵ_2, so this will accomplish the proof.

Assume, therefore, that X is a name of an element of [A]^ℵ_1.
Let τ = ⋂_ζ∈X𝔹_ζ.
Assume toward contradiction that τ is forced to be uncountable.
Fix an element p of ℙ_κ, with the goal to extend p to a condition which forces the opposite statement.
Since p is arbitrary, we will be done.
For a sufficiently large regular χ we fix M≺ℋ(χ) such that p,X,ℙ_κ,τ∈ M and M is countable.
By properness, let q be an (M,ℙ_κ)-generic condition such that p≤ q.
Define:

    σ = {(r,γ)∈ M: r⊩γ̌∈τ}.

The focal point is that q⊩σ=τ∩ M.
We prove the inclusion of τ∩ M in σ, the opposite inclusion being proved exactly in the same manner.
Assume, therefore, that γ∈ M.
For the current direction we assume that there is a condition s≥ q such that s⊩γ̌∈τ.
Since q is (M,ℙ_κ)-generic and the set of conditions which decide the statement γ̌∈τ is dense, one can find t∈ M such that t∥ s ∧ t⊩γ̌∈τ.
By the definition of σ we see that t⊩γ̌∈σ and hence some extension of s forces this statement as well, giving the desired inclusion.
The opposite inclusion is proved with respect to the opposite statement, namely s⊩γ̌∉τ.
Remark that q⊩σ=τ∩ M along with the assumption toward contradiction imply that σ is forced by q to be an infinite set.

Let η=sup(X).
We may assume that this fact is forced by q.
By cutting off the upper part of X if necessary, we may assume that ^V(η)>ω.
Let δ=sup(M∩η).
Recall that M is countable, and hence δ<η.
Choose a condition q^+≥ q such that q^+⊩γ̌∈X∧γ∈[δ,η).
We shall argue that q^+⊩(σ⊆𝔹_γ∩ M), thus arriving at a contradiction.

Our strategy is to prove the above statement in a larger universe, and then using absoluteness in order to show that this holds already in the generic extension by ℙ_κ.
Let 𝕋 be the termspace forcing of ℙ_[η,κ) with respect to ℙ_[δ,η) as computed in the universe obtained by the generic extension with ℙ_δ.
Let ℝ = ℙ_δ∗(ℙ_[δ,η)×𝕋).
Observe that ℝ has a natural projection onto ℙ_κ, namely π(p,q,r)=p^⌢ q^⌢ r.
In particular, there are suitable conditions (a,b,c) such that π(a,b,c)=q^+.
Choose a generic set H for the forcing notion ℙ_δ∗𝕋, such that (a,c)∈ H.
Notice that the name σ is interpreted by H since there are no coordinates of σ in the interval of ℙ_[δ,η) by the definition of σ.

We claim that in V[H] it is true that ∅_ℙ_[δ,η)⊩(σ⊆B_γ∩ M).
Remark that maybe ℵ_1 is collapsed in V[H] (though not in V[G]), but our claim is still valid.
Indeed, if p is a finite condition in the forcing which adds 𝔹_γ then we can extend p by adding a pair which will be forced to be in σ but not in 𝔹_γ∩ M.
For this end, let α=sup(σ).
Since p is finite, there is a maximal pair (ν,ξ) in p.
Choose an ordinal ρ∈σ∩(ξ,α) such that ρ+1 will not be in σ (such an ordinal always exists) and add the pair (ν+1,ρ+1) to p.
This shows that (σ⊆𝔹_γ∩ M) in the generic extension by H.
Since this statement is Δ_0 we conclude that V[G](σ⊆𝔹_γ∩ M) as well, so we are done.

thmpfa

We make the comment that an additional argument shows that Galvin's property fails in a strong sense in the above model, namely the intersection of any family of ℵ_1 Baumgartner clubs is finite.
The important feature of the usual iteration to force PFA that we used is the fact that the set A of names for Baumgartner clubs along the iteration is unbounded in κ.




amsplain


