The spectrum of independence

We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote Spec ( mi f ) . Here mif abbreviates maximal independent family . We show that: Assuming large cardinals, in we


Introduction
We study the set of possible sizes of maximal independent families. Let A be a family of infinite subsets of ω. Following [10] we denote by FF(A) the set of all partial functions h : A → 2 with finite domain, denoted dom (

h). For h ∈ FF(A) let A h = {A h(A) : A ∈ dom(h)}, where A h(A) = A if h(A) = 0 and A h(A) = ω\A if h(A) = 1. A family A ⊆ [ω]
ω is said to be independent if for every h ∈ FF(A), the set A h is infinite. It is maximal independent if in addition, it is not properly included in another maximal independent family. The minimal size of a maximal independent family is denoted i and is referred to as the independence number.
Compared to the other classical cardinal characteristics of the continuum, the independence number seems to be one of the less studied (for an excellent exposition of the subject of cardinal characteristics, we refer the reader to [2]). In this article we study the set of possible sizes of maximal independent families, to which we refer as spectrum of independence and denote Spec(mi f ). It seems surprisingly difficult to control those possible sizes. A Cohen generic real for example, destroys the maximality of all ground model maximal independent families. Below i are d and r (see [2]), as well as cof(M) (see [1]). However apart from c, there are no other known upper bounds. In [10] the second author of the current article shows that consistently i < u = c = ℵ 2 , construction which will later be observed to provide the existence of a Sacks indestructible maximal independent families. For a detailed proof of the existence of such families see [4]. Alternatively the consistency of i < c can be obtained via a finite support iterations of ccc posets (see [6,Proposition 18.11]), result due to Brendle. Recent studies on the structure of independent families can be found in [4,8].
Our article is organized as follows: In Sect. 2, to a given independent family A we associate a family of special filters U, to which we refer as an A-diagonalization filters, such that the relativized Mathias poset M(U) adjoins a generic real σ A with the following diagonalization property: A ∪ {σ A } is independent and furthermore for each x ∈ V ∩ ([ω] ω \A) such that A ∪ {x} is independent, the family A ∪ {x, σ A } is not maximal. This property allows us in an appropriate finite support iteration to guarantee that any finite set of regular cardinals does appear as a subset of Spec(mi f ) (see Theorem 5). In Sect. 3, we study Sacks extensions (extensions obtained via long countable support products of Sacks forcing) of models of CH and show that in those models there are no maximal independent families of intermediate size, i.e. of cardinalities λ where ℵ 1 < λ < c. Finally, in Sect. 4, at the price of assuming large cardinals, we show that the spectrum of independence is not necessarily convex. In fact, the spectrum can exclude finitely many intervals of the form (κ i , κ i+1 ) = {λ : κ i < λ < κ i+1 }. We conclude with some well known open questions, which motivated this work. More is in a paper under preparation.

Diagonalizing an independent family
Let A be an independent family and let bhull(A) be the set of all Boolean combinations of A. That is bhull(A) = {A h : h ∈ FF(A)}. Then the Frechét filter, denoted F 0 , has the following two properties: 1. ∀F ∈ F 0 ∀B ∈ bhull(A), F ∩ B is infinite, and 2. F 0 ∩ bhull(A) = ∅.

Definition 1 Let
A be an independent family. A filter U is said to be an Adiagonalization filter, if U extends F 0 and U is maximal with respect to the above two properties.
Whenever U is a filter, denote by M(U) the Mathias poset relativized to U. The conditions of M(U) are all pairs of the form (s,

Lemma 2 Let
A be an independent family, U an A-diagonalization filter and let G be M(U)-generic filter. Let x G = {s : ∃F(s, F) ∈ G}. Then: Again, fix h, n as above and consider the set Consider an arbitrary (s, F) ∈ M(U). Find an initial segment t of A h \(max s + 1) such that |t| > n. Then (s, F\(max t + 1)) is an extension of (s, F) from E h,n and so E h,n is dense. Again, since h, n were arbitrary we obtain that A h \x G is infinite, for each h.
The above Lemma gives rise to the following more general definition:

Corollary 4 Let A be an independent family, U an A-diagonalization filter and let G be a M(U)-generic filer. Then the Mathias generic real
diagonalizes A over the ground model.
Proceed as follows. If α ∈ I j for some j ∈ {1, . . . , n}, then pick an J j α -diagonalizing filter U α in V P α , takeQ α to be a P α -name for the relativized Mathias poset M(U α ) and r j α to be the associated Mathias generic real. If α / ∈ n j=1 I j , then takeQ α to be a P α -name for the Cohen poset. Now, using standard properties of finite support iteration, the fact that each κ j is regular uncountable and property (2) of Lemma 2, one can easily show that in V P γ * for each j ∈ {1, . . . , n} the family J j = {r j γ : γ ∈ I j } is maximal independent.
For clarity, we presented the proof of the above theorem in case the set of values {κ j } n j=1 we want to guarantee to appear in Spec(mi f ) is finite. However, the above argument clearly generalizes. Let λ be the intended size of the continuum, where cof(λ) > ℵ 0 . Partition λ into θ -many disjoint sets I j : j ∈ θ , such that |I j | = κ j , for some regular uncountable κ j and I j cofinal in λ. Using an appropriate bookkeeping function we can do a finite support iteration, such that the iterands corresponding to I j adjoin a maximal independent family of size κ j . Then in the final generic extension {κ j : j ∈ θ } ⊆ Spec(mi f ).

The spectrum is not necessarily convex
In the following, we will show that the spectrum is not necessarily convex. In fact, it can be rather small, consisting only of ℵ 1 and c. In [10], in a model of CH, the second author constructed a maximal independent family which remains a natural witness to i = ℵ 1 in a generic extension with u = c = ℵ 2 . The construction gives rise to the existence of a Sacks indestructible maximal independent family. That is a maximal independent family, which remains maximal after the countable support iteration of Sacks forcing. A detailed proof of this fact can be found in [4].

Theorem 6 ([4], Corollary 36; [10]) (CH) There is a maximal independent family, which remains maximal after the countable support iteration of Sacks forcing, as well as after an arbitrarily long countable support product of Sacks forcing.
An alternative proof of the above theorem which uses diamond principles can be found in [7]. Proof Fix κ such that ℵ 1 < κ < λ. We will show that if A is an independent family of cardinality κ, then A is not maximal. Towards a contradiction suppose there is p ∈ P and a family {τ α : α < κ} of P-names for subsets of ω such that p ({τ α : α < κ} is max independent). For α < ℵ 2 , let p α ≤ p and let U α ∈ [λ] ℵ 0 be such that the support of p α , dom( p α ) = U α and below p α we can read τ α continuously (for a detailed presentation of continuous reading of names see [9]). 1 Whenever τ is a nice P-name for an infinite subset of ω and p ∈ P, we denote by τ (≤ p) the natural restriction of τ below p. Now, we can find S ∈ [ω 2 ] ℵ 2 such that ( ) U α : α ∈ S is a -system with root U , the sequence otp(U α ) : α ∈ S is constant, and for α = β from S, if π α,β is the order preserving function from U β onto U α , then π α,β U = id U , π α,β maps τ β (≤ p β ) onto τ α (≤ p α ). Now, each τ α is wlog the P-name depending only on We can find U such that U ⊆ λ, otp(U) = otp(U α ) for α ∈ S, U ∩W = U . If α ∈ S let π α, be the order preserving function from U onto U α . Then consider the condition p = π −1 α, ( p α ) and the naturally defined name τ = π −1 (τ α (≤ p α )). Now p ≤ p α and p ({τ } ∪ {τ α : α ∈ κ} is independent), which contradicts p ({τ α : α < κ} is maximal).

Excluding values
Let κ be a measurable cardinal and let D ⊆ P(κ) be a κ-complete ultrafilter. Let P be a partial order. Then P κ /D is defined as the set of all equivalence classes and is supplied with the partial order relation We can identify each p ∈ P with the equivalence class [ p] = [ f p ], where f p (α) = p for each α ∈ κ and so we can assume P ⊆ P κ /D. The following claims can be found in [3, Lemmas 0.1 and 0.2].
Claim 8 1. The poset P is a complete suborder of P κ /D if and only if P is κ-cc. Thus, if P is ccc, then P P κ /D. 2. If P has the countable chain condition, then so does P κ /D.
Taking ultrapowers destroys the maximality of small independent families.

Lemma 9
Let P be a ccc poset and let A be a P-name for an independent family such that P (A is independent). Then Proof We can assume that P |A| = λ ≥ κ. Then, each element A α of A is represented by countably many antichains { p α n,i : i ∈ ω} and {k α n,i } ⊆ {0, 1} such that LetȦ be the average of the namesȦ α for α < κ. That is, for each n, i [ p n,i ] =< p α n,i : α ∈ κ > /D and k n,i =< k α n,i : α < κ > /D.
We claim that P κ /D (A ∪ {Ȧ} is independent). Fix an arbitrary Boolean combination B β of A. Then for all but finitely many α, P B β ∩ A α is infinite. By the theorem of Łoś for the L κ,κ -language we obtain that the average of the A α 's meets B β on an infinite set.
We denote by Even the class of all ordinals α such that α = β + 2k for some limit β and k ∈ ω, and by Odd the class of ordinals α which can be written in the form α = β + 2k + 1 where β is a limit and k ∈ ω.
Theorem 10 Let κ 1 < κ 2 < · · · < κ n be measurable cardinals witnessed by κ icomplete ultrafilters D i ⊆ P(κ i ). Then there is a ccc generic extension in which Proof We will modify the proof of Theorem 5 as follows. Thus, fix γ * and I j ⊆ γ * for each j = 1, . . . , n as in the proof of 5, but assume in addition that I j consists of successor cardinals and γ * = sup{γ ∈ I j : γ ∈ Even} = sup{γ ∈ I j : γ ∈ Odd}. Proceed with the recursive definition of a ccc finite support iteration P α ,Q β : α ≤ γ * , β < γ * . Fix α < γ * and suppose for each j ∈ {1, . . . , n}, we have defined sequences of reals r j γ : γ ∈ I j ∩ Even, γ < α such that J j α = {r j γ : γ ∈ I j ∩ Even ∩ α} is an independent family and for each γ ∈ I j ∩ Even, r j γ diagonalizes J j γ = {r j δ : δ ∈ I j ∩ γ ∩ Even}. Now, continue the construction as follows: If α ∈ I j ∩ Even for some j ∈ {1, . . . , n}, then pick an J j α -diagonalizing filter U α in V P α , takeQ α to be a P α -name for the relativized Mathias poset M(U α ) and r j α to be the associated Mathias generic real. If α ∈ I j ∩ Odd for some j ∈ {1, . . . , n} then α = β + 1 for some β and we can takeQ α to be a P β -name for the quotient poset R β , where P κ j β /D j = P β * R β . If α / ∈ n j=1 I j , then takeQ α to be a P α -name for the Cohen poset.
The reason that each κ i appears in Spec(mi f ) in V P γ * is the same as in Theorem 5. To see that there are no undesired sizes in the spectrum, fix λ such that κ j < λ < κ j+1 for some j ∈ {1, . . . , n − 1} and suppose in V P γ * the family A is independent of cardinality λ. Since P γ * is ccc, we can find α 0 < γ * such that A appears already in V Pα 0 . However I j is cofinal in γ * and we can find an odd α ∈ I j , where α = β + 1 for some β, such that α 0 < β. By Lemma 9 applied to A and P = P β , the family A is not maximal in V P α , and so not maximal in V P γ * .

Concluding remarks
Even though, we just gave an initial analysis of the spectrum of independence our results can be viewed as a very preliminary attempt to address the following two questions: 1. Is it consistent that i < a? Note that the consistency of a < i holds in the random model.
2. Is it consistent that i is of countable cofinality?