MMATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS
DIEGO A. MEJ´IA
Abstract.
We use coherent systems of FS iterations on a power set, which can be seenas matrix iteration that allows restriction on arbitrary subsets of the vertical component,to prove general theorems about preservation of certain type of unbounded families ondefinable structures and of certain mad families (like those added by Hechler’s poset foradding an a.d. family) regardless of the cofinality of their size. In particular, we definea class of posets called σ -Frechet-linked and show that they work well to preserve madfamilies, and unbounded families on ω ω .As applications of this method, we show that a large class of FS iterations can preservethe mad family added by Hechler’s poset (regardless of the cofinality of its size), andthe consistency of a constellation of Cicho´n’s diagram with 7 values where two of thesevalues are singular. Introduction
Background.
In the framework of FS (finite support) iterations of ccc posets to proveconsistency results with large continuum (that is, the size of the continuum c = 2 ℵ largerthan ℵ ), very recently in [FFMM18] appeared the general notion of coherent systems ofFS iterations that was used to construct a three-dimensional array of ccc posets to forcethat the cardinals in Cicho´n’s diagram are separated into 7 different values (see Figure1). This is the first example of a 3D iteration that was used to prove a new consistencyresult. Moreover, the methods from [BF11] where used there to force, in addition, thatthe almost disjointness number a is equal to the bounding number b , and to expandwell-known results about preservation of mad families along FS iterations.For quite some time, consistency results about many different values for cardinal invari-ants has been investigated. Some of the earliest results are due to Brendle [Bre91] whofixed standard techniques for FS iterations in this direction, and due to Blass and Shelah[BS89] who constructed the first example of a two-dimensional array of ccc posets to provethe consistency of the existence of a base for a non-principal ultrafilter in ω of size smallerthan the dominating number d . The latter technique received the name matrix iterations in [BF11] and it was improved there to prove the consistency of e.g. ℵ < b = a < s .Recent developments on matrix iterations appear in work of the author [Mej13a, Mej13b],where forcing models satisfying that several cardinals in Cicho´n’s diagram are pairwisedifferent (at most 6 different values were achieved) are constructed, and of Dow and She-lah [DS18] where the splitting number s is forced to be singular. Concerning Cicho´n’sdiagram, a few months ago Goldstern, Kellner and Shelah [GKS] used Boolean ultra-powers of strongly compact cardinals applied to the iteration constructed in [GMS16] to Mathematics Subject Classification.
Key words and phrases.
Coherent system of finite support iterations, Frechet-linked, preservation ofunbounded families, preservation of mad families, Cicho´n’s diagram.Supported by grant no. IN201711, Direcci´on Operativa de Investigaci´on, Instituci´on UniversitariaPascual Bravo, and by the Grant-in-Aid for Early Career Scientists 18K13448, Japan Society for thePromotion of Science. a r X i v : . [ m a t h . L O ] J u l DIEGO A. MEJ´IA b b b b bb bb b b b b θ θ κ µ ν λ ℵ add( N ) add( M ) cov( M ) non( N ) b d cov( N ) non( M ) cof( M ) cof( N ) c Figure 1. c must be regular. In the context of cccforcing, one of the few exceptions is the consistency of d < c with both cardinals singular,which can be obtained by a FS iteration where the last iterand is a random algebra (seee.g. [FFMM18, Thm. 5.1(d)]). On the other hand, many examples can be obtainedby creature forcing constructions as in [KS09, KS12, FGKS17], for instance, in the latterreference it is proved that the right side of Cicho´n’s diagram can be divided into 5 differentvalues where 4 of them are singular. However, all these constructions are ω ω -bounding,so they force d = ℵ and do not allow separation of cardinal invariants below d . Objective 1.
The main motivation of this research is to improve some of the ccc forcingmethods to produce models where many cardinal invariants of the continuum are differentand two or more of them are singular. As one of the main results of this paper, we showhow to take advantage of the generality of coherent systems of FS iterations to producesuch models where 2 cardinal invariants can be forced to be singular. In particular,we show that the 3D iteration of [FFMM18] that forces the constellation of 7 values inCicho´n’s diagram can be modified so that 2 cardinals are allowed to be singular (Theorem4.3). In addition. we modify examples from [Mej13a, FFMM18] in the same way.
Methods.
A coherent system of FS iterations of length π consists of a partial order h I, ≤i and, for each i ∈ I , a FS iteration P i,π := h P i,α , ˙ Q i,α : α < π i such that any pairof such iterations are coherent in the sense that, whenever i ≤ j in I and α ≤ π , the P i,α -generic extension is contained in the P j,α -generic extension (see details in Definition2.3 and a picture in Figure 6). For instance, a matrix iteration is a coherent system (ofFS iterations) when h I, ≤i is a well-order (see Figure 2), and a 3D iteration is a coherentsystem on a product of ordinals I = γ × δ with the coordinate-wise order (see Figure 3).For our applications, we construct coherent systems on partial orders of the form hP (Ω) , ⊆i , which in fact look like matrix iterations, with vertical component indexedby Ω, that allow restriction on any arbitrary subset of Ω. To be more precise, as the finalgeneric extension of the forcing produced by such a system comes from the FS iteration h P Ω ,ξ , ˙ Q Ω ,ξ : ξ < π i , for any A ⊆ Ω the iteration h P A,ξ , ˙ Q A,ξ : ξ < π i can be understood asthe ‘vertical’ restriction on A of the former FS iteration (see Figure 4). This “restriction” ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 3 bbbbb bbbbb bbbbb bbbbb V , V , V α, V α +1 , V γ, V ,ξ V ,ξ V α,ξ V α +1 ,ξ V γ,ξ V ,ξ +1 V ,ξ +1 V α,ξ +1 V α +1 ,ξ +1 V γ,ξ +1 ˙ Q ,ξ ˙ Q ,ξ ˙ Q α,ξ ˙ Q α +1 ,ξ ˙ Q γ,ξ V ,π V ,π V α,π V α +1 ,π V γ,π Figure 2.
Matrix iteration. bbb bb b b bb b bb b bb b bb b b V , , V ,δ, V α,β, V γ, , V γ,δ, V , ,ξ V , ,ξ +1 V , ,π V ,δ,ξ V ,δ,ξ +1 V ,δ,π V α,β,ξ V α,β,ξ +1 V α,β,π V γ, ,ξ V γ, ,ξ +1 V γ, ,π V γ,δ,ξ V γ,δ,ξ +1 V γ,δ,π ˙ Q , ,ξ ˙ Q ,δ,ξ ˙ Q α,β,ξ ˙ Q γ, ,ξ ˙ Q γ,δ,ξ Figure 3.
Three-dimensional iteration.feature is what allows nice combinatorial arguments to force singular values for some cardi-nal invariants. Concretely, it allows to preserve unbounded reals (with respect to generalstructures, see Definition 3.4) that come from the vertical component (Theorem 3.15),and even maximal almost disjoint (mad) families of size of singular cardinality (Theorem3.32). Surprisingly, the three-dimensional forcings from [FFMM18] can be reconstructednow as matrix iterations with vertical support restriction, though the real picture of thelatter is the ‘shape’ of the lattice hP (Ω) , ⊆i plus one additional dimension (for the FSiterations). Objective 2.
The theory of Brendle and Fischer [BF11] for preserving mad familiesis the cornerstone for the preservation results we propose in this paper, as well as itis in [FFMM18]. In the latter reference it is proved that E (the standard σ -centeredposet adding an eventually different real) and random forcing (thus any random algebra) DIEGO A. MEJ´IA A Ω V P Ω ,π V P A,ξ V P Ω ,ξ V P A,π
Figure 4.
Matrix iteration with vertical support restriction.behaves well in their preservation theory, which allows to prove in [FFMM18, Thm. 4.17]that, whenever κ is an uncountable regular cardinal, the mad family added by the Hechlerposet H κ is preserved by any further FS iteration whose iterands are either E , a randomalgebra or a ccc poset of size < κ . In relation to this, we define a class of posets,which we call σ -Frechet-linked (see Definition 3.24), that includes E and random forcing,and we prove that any (definable) poset in this class behaves well with Brendle’s andFischer’s preservation theory (Theorem 3.27). Moreover, by using coherent systems ona power set hP (Ω) , ⊆i we generalize [FFMM18, Thm. 4.17] by proving that, wheneverΩ is uncountable (not necessarily of regular size), the mad family added by H Ω can bepreserved by a large class of FS iterations (which includes the Suslin σ -Frechet-linkedposets as iterands, see Theorem 4.1). This is related to the preservation of mad familiesof singular size discussed in the previous paragraph. In addition we also show that, for acardinal µ , µ -Frechet-linked posets behave well in the preservation theory of unboundedfamilies (Theorem 3.30). Structure of the paper.
In Section 2 we review the notion of coherent systems of FSiterations and prove general theorems about (vertical) direct limits within such a system.Section 3 is divided in two parts. In the first part, we review Judah’s and Shelah’s [JS90]and Brendle’s [Bre91] theory of preservation of strongly unbounded families (with respectto general definable structures), as well as known facts from [BS89, BF11, Mej13a] topreserve unbounded reals. At the end, a general theorem about preservation of unboundedreals through coherent systems on a power set hP (Ω) , ⊆i is proved. In the second part wereview Brendle’s and Fischer’s theory for mad family preservation, define µ -Frechet-linkedposets and prove that they behave well in this preservation theory. This allows to prove atthe end a general theorem about preservation of mad families through coherent systemson a power set hP (Ω) , ⊆i . Afterwards, in Section 4 we show applications of the theorypresented so far, namely, mad family preservation along a large class of FS iterations andconsistency results about Cicho´n’s diagram.The last section proposes a general framework for linkedness of subsets of posets thatincludes notions like n -linked, centered and Frechet-linked. We say that Γ is a linkedness ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 5 b bb b
MN M [ G ∩ P ] N [ G ] PQ Figure 5.
Generic extensions of pairs of posets ordered like P l M Q . property of subsets of posets if Γ( P ) defines a family of subsets of P for each poset P . Forsuch a property Γ we define its corresponding notions of θ - Γ -Knaster (a poset P has thisproperty iff any subset of size θ contains a subset in Γ( P ) of size θ ) and µ - Γ -covered (theversion of µ -linked for Γ). Built on the classical FS product and iteration theorems forKnaster and σ -linked, we find sufficient conditions for Γ to generalize these theorems for θ -Γ-Knaster and µ -Γ-covered. Some notation.
Denote by C the poset that adds one Cohen real and by C Ω the posetthat adds a family of Cohen reals indexed by the set Ω (which is basically a finite supportproduct of C ). The Lebesgue measure on the Cantor space 2 ω is denoted by Lb. Randomforcing , denoted by B , is the poset of Borel subsets of 2 ω with positive Lebesgue measure,ordered by ⊆ . A random algebra on a set Ω, denoted by B Ω , is the poset of subsets of2 Ω × ω of the form B × (Ω r J ) × ω for some J ⊆ Ω countable and some Borel subset B of2 J × ω with positive Lebesgue measure, ordered by ⊆ . This adds a family of random realsindexed by Ω. Hechler poset for adding a dominating real is denoted by D , and E isdefined as the poset whose conditions are pairs ( s, ϕ ) with s ∈ ω <ω and ϕ : ω → [ ω ] ≤ m for some m < ω , ordered by ( s , ϕ ) ≤ ( s, ϕ ) iff s ⊆ s , ϕ ( i ) ⊆ ϕ ( i ) for any i < ω , and s ( i ) / ∈ ϕ ( i ) for any i ∈ | s | r | s | . The trivial poset is denoted by .Most of the cardinal invariants used in this paper are defined (or characterized) inExample 3.7. Recall that A ⊆ [ ω ] ℵ is an almost disjoint (a.d.) family if the intersectionof any two different members of A is finite. A mad family is a maximal a.d. family, and a is defined as the smallest size of an infinite mad family. For a set Ω, Hechler’s poset H Ω for adding an a.d. family (indexed by Ω) is defined as the poset whose conditions are ofthe form p : F p × n p → F p ∈ [Ω] <ω and n p < ω (demand n p = 0 iff F p = ∅ ), orderedby q ≤ p iff p ⊆ q and | q − [ { } ] ∩ ( F p × { i } ) | ≤ i ∈ [ n p , n q ) (see [Hec72]).This poset has the Knaster property and the a.d. family it adds is maximal when Ω isuncountable. It is forcing equivalent to C when Ω is countable and non-empty, and it isequivalent to C ω when | Ω | = ℵ . For any Ω ⊆ Ω , H Ω l H Ω .2. Coherent systems of FS iterations
Definition 2.1.
Let M be a transitive model of ZFC. When P ∈ M and Q are posets,say that P is a complete subposet of Q with respect to M , abbreviated P l M Q , if P isa subposet of Q and any maximal antichain of P that belongs to M is still a maximalantichain in Q .If in addition N is another transitive model of ZFC, M ⊆ N and Q ∈ N , then P l M Q implies that, whenever G is Q -generic over N , G ∩ P is P -generic over M and M [ G ∩ P ] ⊆ N [ G ] (see Figure 5). Example 2.2.
Let M ⊆ N be transitive models of ZFC. When P ∈ M it is clear that l M P and P l M P . Also, if S is a Suslin ccc poset or a random algebra coded in M then S M l M S N . DIEGO A. MEJ´IA bbb bb bbb b b bb b bb b bb b bb b bb b bb b b
V V i, V j, V i,ξ V i,ξ +1 V i,π V j,ξ V j,ξ +1 V j,π ˙ Q i,ξ ˙ Q j,ξ Figure 6.
Coherent system of FS iterations. The figures in dashed linesrepresent the ‘shape’ of the partial order h I, ≤i . Definition 2.3 ([FFMM18, Def. 3.2]) . A coherent system (of FS iterations) s is composedof the following objects:(I) a partially ordered set I s , an ordinal π s , and(II) for each i ∈ I s , a FS iteration P s i,π s = h P s i,ξ , ˙ Q s i,ξ : ξ < π s i such that, for any i ≤ j in I and ξ < π s , if P s i,ξ l P s j,ξ then P s j,ξ forces ˙ Q s i,ξ l V P s i,ξ ˙ Q s j,ξ .According to this notation, P s i, is the trivial poset and P s i, = ˙ Q s i, . We often refer to h P s i, : i ∈ I s i as the base of the coherent system s . The condition given in (II) impliesthat P s i,ξ l P s j,ξ whenever i ≤ j in I s and ξ ≤ π s (see Lemma 3.14).For j ∈ I s and η ≤ π s we write V s j,η for the P s j,η -generic extensions. Concretely, when G is P s j,η -generic over V , V s j,η := V [ G ] and V s i,ξ := V [ P s i,ξ ∩ G ] for all i ≤ j in I s and ξ ≤ η .Note that V s i,ξ ⊆ V s j,η and V s i, = V (see Figure 6).We say that the coherent system s has the ccc if, additionally, P s i,ξ forces that ˙ Q s i,ξ hasthe ccc for each i ∈ I s and ξ < π s . This implies that P s i,ξ has the ccc for all i ∈ I s and ξ ≤ π s .A concrete simple type of coherent system is what we call a coherent pair (of FSiterations) . A coherent system s is a coherent pair if I s is of the form { i , i } ordered as i < i .For a coherent system s and a set J ⊆ I s , s | J denotes the coherent system with I s | J = J , π s | J = π s and the FS iterations corresponding to (II) defined as for s ; if η ≤ π s , s (cid:22) η denotesthe coherent system with I s (cid:22) η = I s , π s (cid:22) η = η and the iterations for (II) defined up to η asfor s . Note that, if i < i in I s , then s |{ i , i } is a coherent pair and s |{ i } is just the FSiteration P s i,π s = h P s i,ξ , ˙ Q s i,ξ : ξ < π s i .In particular, the upper indices s are omitted when there is no risk of ambiguity.The following is a generalization of [FFMM18, Lemma 3.7]. Lemma 2.4.
Let θ be an uncountable regular cardinal. Assume that s is a coherent systemthat satisfies: ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 7 (i) I has a maximum i ∗ and I r { i ∗ } is < θ -directed,(ii) each P i,ξ forces that ˙ Q i,ξ is θ -cc, and(iii) for any ξ < π , if P i ∗ ,ξ is the direct limit of h P i,ξ : i < i ∗ i then P i ∗ ,ξ forces that ˙ Q i ∗ ,ξ = S i
Let s be a standard coherent system and let θ be an uncountable regularcardinal. If(i) I has a maximum i ∗ , I r { i ∗ } is < θ -directed,(ii) i ∗ / ∈ ran∆ , and(iii) whenever π > , P i ∗ , is the direct limit of h P i, : i < i ∗ i ,then (a) and (b) of Lemma 2.4 hold.Proof. It is clear that hypotheses (i)-(iii) of Lemma 2.4 are satisfied. (cid:3)
In our applications we use coherent systems on a power set hP (Ω) , ⊆i . If s is a standardcoherent system based in such a partial order, then P Ω ,π (the largest poset in the system)can be represented as a two dimensional forcing construction supported in the plane Ω × π and, for any A ⊆ γ and ξ ≤ π , P A,ξ is seen as the restriction of the construction to therectangle A × ξ (see Figure 4 in Section 1). The following result is a suitable consequenceof Corollary 2.6 when dealing with such (standard) coherent systems. Lemma 2.7.
Let θ be a cardinal of uncountable cofinality and let s be a standard coherentsystem where I = I s is a suborder of hP (Ω) , ⊆i . Assume that(i) I is closed under intersections,(ii) I ∩ [Ω] <θ is cofinal in [Ω] <θ ,(iii) ∆( ξ ) ∈ [Ω] <θ for any ξ ∈ [1 , π ) (see Definition 2.5(I)(ii)), and(iv) whenever π > and X ∈ I , P X, is the direct limit of h P A, : A ∈ I ∩ [ X ] <θ i .Then, for every X ∈ I and ξ ≤ π ,(a) P X,ξ is the direct limit of h P A,ξ : A ∈ I ∩ [ X ] <θ i and(b) for any P X,ξ -name of a function ˙ x with domain γ < cf( θ ) into S A ∈ I ∩ [ X ] <θ V A,ξ , thereis some A ∈ I ∩ [ X ] <θ such that ˙ x is (forced to be equal to) a P A,ξ -name.
ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 9
Proof.
Fix X ∈ I . The lemma is trivial when | X | < θ , so assume that | X | ≥ θ . By(i) and (ii), I ∩ [ X ] <θ is cofinal in [ X ] <θ . Put I ∗ = I ∩ ([ X ] <θ ∪ { X } ). Hence X is themaximum of I ∗ and I ∗ r { X } = I ∩ [ X ] <θ is < cf( θ )-directed by (i) and (ii). Consider s ∗ = s | I ∗ . Note that s ∗ is a standard coherent system similar to s with the differencethat ∆ ∗ = ∆ s ∗ : [1 , π ) → I ∩ [ X ] <θ is defined as ∆ ∗ ( ξ ) := ∆( ξ ) whenever ∆( ξ ) ⊆ X , or∆ ∗ ( ξ ) := ∅ otherwise. Also, ˙ S s ∗ ξ = ˙ S s ξ and ˙ Q s ∗ ξ = ˙ Q s ξ in the first case, otherwise each oneis the trivial poset. As X / ∈ ran∆ ∗ , the result is a direct consequence of Corollary 2.6applied to s ∗ and cf( θ ). (cid:3) Example 2.8.
Let Ω be a set.(1) The partial order hP (Ω) , ⊆i clearly satisfies conditions (i) and (ii) of Lemma 2.7 (forany infinite cardinal θ ).Assume that Ω = Ω ∪ Ω is a disjoint union. If s is a standard coherent system on hP (Ω) , ⊆i such that P X, = H X ∩ Ω × C X ∩ Ω for any X ⊆ Ω, then condition (iv) ofLemma 2.7 is satisfied for θ = ℵ (and hence for any uncountable θ ).(2) If θ is a regular cardinal, Ω = θ and I is a cofinal subset of θ , then I := I ∪ { θ } satisfies conditions (i) and (ii) of Lemma 2.7. Such a partial order I (in particular I = θ ∪ { θ } ) is used to construct classical matrix iterations as in, e.g., [BS89, BF11,Mej13a, FFMM18]. 3. Preservation properties
As mentioned in the introduction, this section is divided in two parts. For conveniencewith the notation fixed in the first part, we use a different notation from [BF11, FFMM18]for the results in the second part.3.1.
Preservation theory.
A generalization of the contents of this part, as well as com-plete proofs and more examples, can be found in [CM, Sect. 4].Typically, cardinal invariants of the continuum are defined through relational systems as follows.
Definition 3.1. A relational system is a triplet R = h X, Y, @ i where @ is a relationcontained in X × Y . For x ∈ X and y ∈ Y , x @ y is often read y @ -dominates x . Afamily F ⊆ X is R -unbounded if there is no real in Y that @ -dominates every member of F . Dually, D ⊆ Y is an R -dominating family if every member of X is @ -dominated bysome member of D . The cardinal b ( R ) denotes the least size of an R -unbounded familyand d ( R ) is the least size of an R -dominating family.Say that x ∈ X is R -unbounded over a set M if x @ y for all y ∈ Y ∩ M . Given acardinal λ say that F ⊆ X is λ - R -unbounded if, for any Z ⊆ Y of size < λ , there isan x ∈ F that is R -unbounded over Z . Say that F ⊆ X is strongly λ - R -unbounded if | F | ≥ λ and |{ x ∈ F : x @ y }| < λ for any y ∈ Y . Remark 3.2.
When λ ≥
2, any λ - R -unbounded family is R -unbounded. Hence, if F isa λ - R -unbounded family then b ( R ) ≤ | F | and λ ≤ d ( R ). Also, if θ is regular and F is astrongly θ - R -unbounded family then it is | F | - R -unbounded, so b ( R ) ≤ | F | ≤ d ( R ). Definition 3.3.
Let R = h X, Y, @ i and R = h X , Y , @ i be two relational systems. Saythat R is Tukey-Galois below R if there are two maps F : X → X and G : Y → Y suchthat, for each x ∈ X and b ∈ Y , if F ( x ) @ b then x @ G ( b ). When, in addition, R isTukey-Galois below R , we say that R and R are Tukey-Galois equivalent .Recall that, whenever R is Tukey-Galois below R , b ( R ) ≤ b ( R ) and d ( R ) ≤ d ( R ). Definition 3.4.
A relational system R := h X, Y, @ i is a Polish relational system (Prs) ifthe following is satisfied:(i) X is a perfect Polish space,(ii) Y is a non-empty analytic subspace of some Polish space Z and(iii) @ = S n<ω @ n for some increasing sequence h @ n i n<ω of closed subsets of X × Z suchthat ( @ n ) y = { x ∈ X : x @ n y } is nwd (nowhere dense) for all y ∈ Y .By (iii), h X, M ( X ) , ∈i is Tukey-Galois below R where M ( X ) denotes the σ -ideal ofmeager subsets of X . Therefore, b ( R ) ≤ non( M ) and cov( M ) ≤ d ( R ). Moreover, (iii)implies that, whenever c ∈ X is a Cohen real over a transitive model M of ZFC and thePrs R is coded in M , c is R -unbounded over M . Definition 3.5 (Judah and Shelah [JS90]) . Let R = h X, Y, @ i be a Prs and let θ be acardinal. A poset P is θ - R -good if, for any P -name ˙ h for a real in Y , there is a non-empty H ⊆ Y of size < θ such that (cid:13) x @ ˙ h for any x ∈ X (in the ground model) that is R -unbounded over H .Say that P is R -good when it is ℵ - R -good.Definition 3.5 describes a property, respected by FS iterations, to preserve specific typesof R -unbounded families. Concretely, when θ is uncountable regular,(a) any θ - R -good poset preserves all the (strongly) θ - R -unbounded families from theground model and(b) FS iterations of θ -cc θ - R -good posets produce θ - R -good posets.By Remark 3.2, posets that are θ - R -good work to preserve b ( R ) small and d ( R ) large.Clearly, θ - R -good implies θ - R -good whenever θ ≤ θ , and any poset completely em-bedded into a θ - R -good poset is also θ - R -good.Consider the following particular cases of interest for our applications. Lemma 3.6 ([Mej13a, Lemma 4]) . If R is a Prs and θ is an uncountable regular cardinal,then any poset of size < θ is θ - R -good. In particular, Cohen forcing is R -good. Example 3.7.
Fix an uncountable regular cardinal θ .(1) Preserving non-meager sets:
Consider the Polish relational system Ed := h ω ω , ω ω , = ∗ i where x = ∗ y iff x and y are eventually different, that is, x ( i ) = y ( i ) for all butfinitely many i < ω . By [BJ95, Thm. 2.4.1 and 2.4.7], b ( Ed ) = non( M ) and d ( Ed ) = cov( M ).(2) Preserving unbounded families:
Let D := h ω ω , ω ω , ≤ ∗ i be the Polish relational systemwhere x ≤ ∗ y iff x ( i ) ≤ y ( i ) for all but finitely many i < ω . Clearly, b ( D ) = b and d ( D ) = d .Miller [Mil81] proved that E is D -good. Furthermore, ω ω -bounding posets, like therandom algebra, are D -good. In Theorem 3.30 we prove that µ -Frechet-linked posetsare µ + - D -good.(3) Preserving null-covering families:
Define X n := { a ∈ [2 <ω ] < ℵ : Lb( S s ∈ a [ s ]) ≤ − n } (endowed with the discrete topology) and put X := Q n<ω X n with the product topol-ogy, which is a perfect Polish space. For every x ∈ X denote N ∗ x := T n<ω S s ∈ x n [ s ],which is clearly a Borel null set in 2 ω .Define the Prs Cn := h X, ω , @ i where x @ z iff z / ∈ N ∗ x . Recall that any null setin 2 ω is a subset of N ∗ x for some x ∈ X , so Cn and hN (2 ω ) , ω , are Tukey-Galoisequivalent. Therefore b ( Cn ) = cov( N ) and d ( Cn ) = non( N ).By a similar argument as in [Bre91, Lemma 1 ∗ ], any ν -centered poset is θ - Cn -goodfor any ν < θ infinite. In particular, σ -centered posets are Cn -good. ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 11 (4)
Preserving “union of null sets is not null”:
For each k < ω let id k : ω → ω such thatid k ( i ) = i k for all i < ω and put H := { id k +1 : k < ω } . Let Lc := h ω ω , S ( ω, H ) , ∈ ∗ i be the Polish relational system where S ( ω, H ) := { ϕ : ω → [ ω ] < ℵ : ∃ h ∈ H∀ i < ω ( | ϕ ( i ) | ≤ h ( i )) } , and x ∈ ∗ ϕ iff ∃ n < ω ∀ i ≥ n ( x ( i ) ∈ ϕ ( i )), which is read x is localized by ϕ . As aconsequence of Bartoszy´nski’s characterization (see [BJ95, Thm. 2.3.9]), b ( Lc ) =add( N ) and d ( Lc ) = cof( N ).Any ν -centered poset is θ - Lc -good for any ν < θ infinite (see [JS90]) so, in partic-ular, σ -centered posets are Lc -good. Moreover, Kamburelis [Kam89] proved that anyBoolean algebra with a strictly positive finitely additive measure is Lc -good. As aconsequence, subalgebras (not necessarily complete) of random forcing are Lc -good. Lemma 3.8. If R = h X, Y, @ i is a Prs then, for any set Ω , H Ω is R -good.Proof. Let ˙ y be a H Ω -name of a member of Y . Then there is some countable A ⊆ Ω suchthat ˙ y is a H A -name. As H A is countable, it is R -good by Lemma 3.6, so there is somenon-empty countable H ⊆ Y witnessing this. The same H witnesses goodness for ˙ y and H Ω . (cid:3) In a similar way, it can be proved that any random algebra is R -good iff random forcingis R -good.The following results indicate that (strongly) ν -unbounded families can be added withCohen reals, and the effect on b ( R ) and d ( R ) by a FS iteration of good posets. Lemma 3.9.
Let ν be a cardinal of uncountable cofinality, R = h X, Y, @ i a Prs and let h P α i α<ν be a l -increasing sequence of cf( ν ) -cc posets such that P ν = limdir α<ν P α . If P α +1 adds a Cohen real ˙ c α ∈ X over V P α for any α < ν , then P ν forces that { ˙ c α : α < ν } is a strongly ν - R -unbounded family of size ν . Theorem 3.10.
Let θ be an uncountable regular cardinal, R = h X, Y, @ i a Prs, δ ≥ θ anordinal and let h P α , ˙ Q α i α<δ be a FS iteration of non-trivial θ - R -good θ -cc posets. Then, P δ forces b ( R ) ≤ θ and d ( R ) ≥ | δ | .Proof. See e.g. [CM, Thm. 4.15] or [GMS16, Cor. 3.6]. (cid:3)
Fix transitive models M ⊆ N of ZFC and a Polish relational system R = h X, Y, @ i coded in M . The following results are related to preservation of R -unbounded reals alongcoherent pairs of FS iterations. Lemma 3.11 ([Mej13a, Thm. 7]) . Let S be a Suslin ccc poset coded in M . If M | = “ S is R -good” then, in N , S N forces that every real in X ∩ N that is R -unbounded over M isstill R -unbounded over M S M . Corollary 3.12.
Let Γ ∈ M be a non-empty set. If M | = “ B Γ is R -good” then B N Γ , in N , forces that every real in X ∩ N that is R -unbounded over M is still R -unbounded over M B M Γ . Lemma 3.13 ([BF11, Lemma 11], see also [Mej15, Lemma 5.13]) . Assume P ∈ M is aposet. Then, in N , P forces that every real in X ∩ N that is R -unbounded over M is still R -unbounded over M P . Lemma 3.14 (Blass and Shelah [BS89], [BF11, Lemmas 10, 12 and 13]) . Let s be acoherent pair of FS iterations (wlog I s = { , } ). Then, P ,ξ l P ,ξ for all ξ ≤ π . Moreover, if ˙ c is a P , -name of a real in X , π is limit and P ,ξ forces that ˙ c is R -unbounded over V ,ξ for all < ξ < π , then P ,π forces that ˙ c is R -unbounded over V ,π . We finish this part with the main result of this subsection.
Theorem 3.15.
Let R = h X, Y, @ i be a Prs, θ a cardinal of uncountable cofinality andlet s be a standard coherent system of FS iterations of length π > that satisfies thehypothesis of Lemma 2.7. Further assume that(i) Γ ⊆ Ω has size ≥ θ ,(ii) D ∈ I and Γ ⊆ D ,(iii) for each l ∈ Γ , P D, adds a real ˙ c l in X such that, for any A ⊆ D in I ∩ [Ω] <θ , if l ∈ D r A then P D, forces that ˙ c l is R -unbounded over V A, , and(iv) for every ξ ∈ S s and B ∈ I ∩ [Ω] <θ , P B,ξ forces that ˙ Q B,ξ is R -good.Then P D,π forces that the family ˙ F := { ˙ c l : l ∈ Γ } is strongly θ - R -unbounded. Inparticular, P D,π forces b ( R ) ≤ | ˙ F | and, when θ is regular, this poset forces | ˙ F | ≤ d ( R ) .Proof. Let ˙ y be a P D,π -name of a member of Y . By Lemma 2.7, there is some A ∈ I ∩ [ D ] <θ such that ˙ y is a P A,π -name. Fix l ∈ Γ r A . By Lemmas 3.11, 3.13, 3.14 and Corollary3.12 applied to the coherent pair s |{ A, D } , P D,π forces that ˙ c l is R -unbounded over V A,π ,which implies that ˙ c l @ ˙ y . Therefore, P D,π forces that { x ∈ ˙ F : x @ ˙ y } ⊆ { ˙ c l : l ∈ Γ ∩ A } ,which has size ≤ | A | < θ .The second statement is a consequence of Remark 3.2 (cid:3) Preservation of mad families.Definition 3.16.
Fix A ⊆ [ ω ] ℵ .(1) Let P ⊆ (cid:2) [ ω ] ℵ (cid:3) < ℵ . For x ⊆ ω and h : ω × P → ω , define x @ ∗ h by ∀ ∞ n < ω ∀ F ∈ P ([ n, h ( n, F )) r [ F * x ) . (2) Define the relational system Md ( A ) := h [ ω ] ℵ , ω ω × [ A ] < ℵ , @ ∗ i .(3) Say that a poset P is uniformly Md ( A ) -good if, for any P -name ˙ h of a member of ω ω × [ A ] < ℵ , there is a non-empty countable H ⊆ ω ω × [ A ] < ℵ (in the ground model) suchthat, for any countable C ⊆ A and any x ∈ [ ω ] ℵ , if x @ ∗ h (cid:22) ( ω × [ C ] < ℵ ) for all h ∈ H then (cid:13) x @ ∗ ˙ h (cid:22) ( ω × [ C ] < ℵ ).Throughout this subsection, fix transitive models M ⊆ N of ZFC and A ∈ M suchthat A ⊆ [ ω ] ℵ ∩ M . The relational system Md ( A ) helps to abbreviate the main notionpresented in [BF11] for the preservation of mad families. What is defined in [BF11, Def. 2]as ( ? M,NA,a ) for a ∈ [ ω ] ℵ , which is the same as “ a diagonalizes M outside A ” in [FFMM18,Def. 4.2], actually means in our notation that a is Md ( A )-unbounded over M . Note that,for any countable C ⊆ [ ω ] ℵ , Md ( C ) is a Prs.The following results from [BF11] indicate that the a.d. family added by H Ω is composedof unbounded reals in the sense of relational systems like in Definition 3.16(2), which inturn becomes a mad family when Ω is uncountable. Lemma 3.17 ([BF11, Lemma 3]) . If a ∗ ∈ [ ω ] ℵ is Md ( A ) -unbounded over M then | a ∗ ∩ x | = ℵ for any x ∈ M r I ( A ) where I ( A ) := { x ⊆ ω : ∃ F ∈ [ A ] < ℵ ( x ⊆ ∗ S F ) } . Lemma 3.18 ([BF11, Lemma 4]) . Let Ω be a set, z ∗ ∈ Ω and ˙ A := h ˙ a z : z ∈ Ω i thea.d. family added by H Ω . Then, H Ω forces that ˙ a z ∗ is Md ( ˙ A (cid:22) (Ω r { z ∗ } )) -unbounded over V H Ω r { z ∗} . ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 13
The known results about the preservation of Md ( A )-unbounded reals along coherentpairs of FS iterations are referred below. This is similar to the previous discussion aboutpreservation of R -unbounded reals for a Prs R . Lemma 3.19 ([BF11, Lemma 12]) . Let s be a coherent pair of FS iterations (wlog I s = { , } ) with π = π s limit, ˙ A a P , -name of a family of infinite subsets of ω and ˙ a ∗ a P , -name for an infinite subset of ω such that (cid:13) P ,ξ “ ˙ a ∗ is Md ( A ) -unbounded over V ,ξ ”for all < ξ < π . Then, P ,π l P ,π and (cid:13) P ,π “ ˙ a ∗ is Md ( A ) -unbounded over V ,π ”. Lemma 3.20 ([BF11, Lemma 11]) . Let P ∈ M be a poset. If N | = “ a ∗ is Md ( A ) -unbounded over M ” then N P | = “ a ∗ is Md ( A ) -unbounded over M P ” . Corollary 3.21. If Ω ∈ M and N | = “ a ∗ is Md ( A ) -unbounded over M ” then N C Ω | = “ a ∗ is Md ( A ) -unbounded over M C Ω ” . Likewise, H Ω satisfies a similar statement. Lemma 3.22 ([FFMM18, Lemma 4.8 and Cor. 4.11]) . Let S be either E or a randomalgebra. If N | = “ a ∗ is Md ( A ) -unbounded over M ” then N S N | = “ a ∗ is Md ( A ) -unbounded over M S M ” . The previous result indicates that E and random forcing, when used as iterands in acoherent pair of FS iterations, help to preserve Md ( A )-unbounded reals. To generalizethis fact, we use the notion of ‘uniformly good’ introduced in Definition 3.16(3). Theorem 3.23.
Let S be a Suslin ccc poset coded in M and A ∈ M , A ⊆ [ ω ] ℵ . Assume ( ? ) [ A ] ℵ ∩ M is cofinal in [ A ] ℵ ∩ N .If M | = “ S is uniformly Md ( A ) -good” then, in N , S N forces that every real in [ ω ] ℵ ∩ N that is Md ( A ) -unbounded over M is still Md ( A ) -unbounded over M S M . Note that, when N is a generic extension of M by a proper poset, ( ? ) holds. Proof.
Let a ∈ [ ω ] ℵ ∩ N be Md ( A )-unbounded over M . Assume that ˙ h ∈ M is a S M -name of a function in ω ω × [ A ] <ω . As S is uniformly Md ( A )-good in M , there is a family { h n : n < ω } ⊆ ω ω × [ A ] < ℵ (in M ) that witnesses goodness for ˙ h . Thus a @ ∗ h n for every n < ω , so we can find a countable C ⊆ A such that a @ ∗ h n (cid:22) ( ω × [ C ] < ℵ ) for every n < ω .By ( ? ), wlog we can find such C in M .In M the statement “for every x ∈ [ ω ] ℵ , if x @ ∗ h n (cid:22) ( ω × [ C ] < ℵ ) for all n < ω , then (cid:13) S x @ ∗ ˙ h (cid:22) ( ω × [ C ] < ℵ )” is true. Furthermore, as this statement is a conjunction of a Σ -statement with a Π -statement of the reals (see e.g. [CM, Claim 4.27]), it is also true in N . In particular, since a @ ∗ h n (cid:22) ( ω × [ C ] < ℵ ) for every n < ω , (cid:13) N S N a @ ∗ ˙ h (cid:22) ( ω × [ C ] < ℵ ). (cid:3) Though E and B are indeed uniformly Md ( A )-good (by Theorem 3.27 and Lemma3.29), the application of Theorem 3.23 yields a version of Lemma 3.22 restricted to thecondition ( ? ). To avoid this restriction, we consider an alternative generalization basedon the following notion. Definition 3.24.
Let P be a poset. (1) Say that a set Q ⊆ P is Frechet-linked (in P ) , abbreviated Fr -linked , if, for anysequence ¯ p = h p n : n < ω i in Q , there is some q ∈ P that forces ∃ ∞ n < ω ( p n ∈ ˙ G ).(2) Let µ be an infinite cardinal. Say that a poset P is µ -Frechet-linked (often abbreviated µ - Fr -linked ) if there is a sequence h Q α : α < µ i of Fr-linked subsets of P such that S α<µ Q α is dense in P .By σ -Fr-linked we mean ℵ -Fr-linked.(3) A poset S is Suslin σ -Frechet-linked if S is a subset of some Polish space, the relations ≤ and ⊥ are Σ (in that Polish space) and S = S n<ω Q n where each Q n is a Fr-linked Σ set.Here, Fr denotes the Frechet filter on ω . The reason of the terminology ‘Frechet-linked’is that this notion corresponds to a particular case on Fr of a more general notion oflinkedness with filters that we provide in Example 5.4. Remark 3.25. (1) The notion ‘ µ -Fr-linked’ is a forcing property, i.e., if P and Q areposets, P l Q (in the sense that the Boolean completion of P is completely embeddedinto the completion of Q ) and Q is µ -Fr-linked then so is P (see more on this in Section5).(2) No Fr-linked subset of a poset can contain infinite antichains. In addition, if P is aposet and Q ⊆ P , the statement “ Q does not contain infinite antichains” is absolutefor transitive models of ZFC. This is because that statement is equivalent to say that“ T is a well-founded tree” where T := { s ∈ Q <ω : ran s is an antichain } .(3) As a consequence of (2), µ -Fr-linked posets are µ + -cc. Even more, by [HT48, Thm.2.4], they are µ + -Knaster (see more in Section 5).(4) Any poset of size ≤ µ is µ -Fr-linked (witnessed by its singletons). In particular, Cohenforcing is σ -Fr-linked.(5) By (3), any Suslin σ -Fr-linked poset is Suslin ccc. Moreover, if h Q n : n < ω i witnessesthat a poset S is Suslin σ -Fr-linked then the statement “ Q n is Fr-linked” is Π (by(6) below, its negation is equivalent to ∃ f ∈ Q ωn ∃ g ∈ S ω ( { g ( n ) : n < ω } is a maximalantichain and ∀ n < ω ∃ m < ω ∀ k ≥ m ( g ( n ) ⊥ f ( k ))), which is Σ ). Therefore, if M | =“ S is Suslin σ -Fr-linked” and ω N ⊆ M then N | =“ S is Suslin σ -Fr-linked”.(6) Let P be a poset and Q ⊆ P . Note that a sequence h p n : n < ω i in Q witnesses that Q is not Fr-linked iff the set { q ∈ P : ∀ ∞ n < ω ( q ⊥ p n ) } is dense. Lemma 3.26.
Let P be a poset and Q ⊆ P Fr -linked. If ˙ n is a P -name of a naturalnumber then there is some m < ω such that ∀ p ∈ Q ( p m < ˙ n ) .Proof. Towards a contradiction, assume that for each m < ω there is some p m ∈ Q thatforces m < ˙ n . Hence, as Q is Fr-linked, there is some q ∈ Q that forces ∃ ∞ m < ω ( p m ∈ ˙ G ),which implies ∃ ∞ m < ω ( m < ˙ n ), a contradiction. (cid:3) Theorem 3.27.
Any σ - Fr -linked poset is uniformly Md ( B ) -good for any B ⊆ [ ω ] ℵ .Proof. Let P be a σ -Fr-linked poset witnessed by h Q n : n < ω i . Assume that ˙ h is a P -name of a function in ω ω × [ B ] <ω . Fix n < ω . For each k < ω and F ∈ [ B ] < ℵ , by Lemma3.26 there is some h n ( k, F ) < ω such that ∀ p ∈ Q n ( p h n ( k, F ) < ˙ h ( k, F )). This allowsto define h n ∈ ω ω × [ B ] <ω . ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 15
Assume that C ⊆ B is infinite and that x ∈ [ ω ] ℵ is Md ( C )-unbounded over { h n (cid:22) ( ω × [ C ] < ℵ ) : n < ω } . We show that (cid:13) x @ ∗ ˙ h (cid:22) ( ω × [ C ] < ℵ ). Assume that p ∈ P and m < ω . Wlog, we may assume that p ∈ Q n for some n < ω . As x @ ∗ h n (cid:22) ( ω × [ C ] < ℵ ),there are some k > m and F ∈ [ C ] < ℵ such that [ k, h n ( k, F )) r S F ⊆ x . On the otherhand, by the definition of h n , there is some q ≤ p in P that forces ˙ h ( k, F ) ≤ h n ( k, F ).Hence q (cid:13) [ k, ˙ h ( k, F )) r S F ⊆ x . (cid:3) As a consequence, Theorem 3.23 can be applied to Suslin σ -Fr-linked posets. However,it can be proved directly that Suslin σ -Fr-linked posets preserves Md ( A )-unbounded realswithout the condition ( ? ). Theorem 3.28.
Let S be a Suslin σ - Fr -linked poset coded in M . Then, in N , S N forcesthat every real in [ ω ] ℵ ∩ N that is Md ( A ) -unbounded over M is still Md ( A ) -unboundedover M S M .Proof. Let h Q n : n < ω i ∈ M be a sequence that witnesses that S is Suslin σ -Fr-linked.Let a ∈ [ ω ] ℵ ∩ N be Md ( A )-unbounded over M . Assume that ˙ h ∈ M is a S M -name of afunction in ω ω × [ A ] <ω . Fix p ∈ S N and m < ω , so there is some n < ω such that p ∈ Q Nn .Down in M , find h n ∈ ω ω × [ A ] <ω ∩ M as in the proof of Theorem 3.27. As a @ ∗ h n , thereare some k ≥ m and F ∈ [ A ] < ℵ such that [ k, h n ( k, F )) r S F ⊆ a . On the other hand,the statement “no q ∈ Q n forces that h n ( k, F ) < ˙ h ( k, F )” is Π , so it is absolute and,as true in M , it also holds in N . Therefore, in N , p does not force h n ( k, F ) < ˙ h ( k, F ),which implies that some q ≤ p in S N forces the contrary. This clearly implies that q forces[ k, ˙ h ( k, F )) r S F ⊆ a . (cid:3) Therefore, in conjunction with the following result, the previous theorem is a suitablegeneralization of Lemma 3.22.
Lemma 3.29.
The posets E and B are Suslin σ - Fr -linked. Moreover, any completeBoolean algebra that admits a strictly positive σ -additive measure (e.g. any random alge-bra) is σ - Fr -linked.Proof. For each s ∈ ω <ω and m < ω define E s,m := { ( t, ϕ ) ∈ E : t = s and ∀ i < ω ( | ϕ ( i ) | ≤ m ) } . This set is actually Borel in ω <ω × P ( ω ) ω (the Polish space where E is defined).A compactness argument similar to the one in [Mil81] shows that E s,n is Fr-linked in E .Fix a non-principal ultrafilter U on ω and let h p n : n < ω i be a sequence in E s,m . Write p n = ( s, ϕ n ). For each i < ω define ϕ ( i ) ⊆ ω such that l ∈ ϕ ( i ) iff { n < ω : l ∈ ϕ n ( i ) } ∈ U .It can be proved that | ϕ ( i ) | ≤ m , (so q := ( s, ϕ ) ∈ E ) and that q forces ∃ ∞ n < ω ( p n ∈ ˙ G )(that is, for any q ≤ q and n < ω , there is some k ≥ n such that q is compatible with p k ).Now, consider random forcing as B = S m<ω B m where( ) B m := n T ⊆ <ω : T is a well-pruned tree and Lb([ T ]) ≥ m + 1 o Note that B m is Borel in 2 <ω . It is enough to show that B m is Fr-linked. Assume thecontrary, so by Remark 3.25(6) there are a sequence h T n : n < ω i in B m and a partition h A n : n < ω i of 2 ω into Borel sets of positive measure such that, for each n < ω , A n ∩ [ T k ]has measure zero for all but finitely many k < ω . Construct an increasing function g : ω → ω such that A n ∩ [ T k ] has measure zero for all k ≥ g ( n ). As 2 ω = S n<ω A n , we A well-pruned tree is a non-empty tree such that every node has a successor. can find some n ∗ < ω such that the measure of A ∗ := S n There is a Suslin σ -Fr-linked poset that is not Lc -good. For b, h ∈ ω ω such that ∀ i < ω ( b ( i ) > 0) and h goes to infinity, consider the poset LOC b,h := (cid:26) p ∈ Y i<ω [ b ( i )] ≤ h ( i ) : ∃ m < ω ∀ ∞ i < ω ( | p ( i ) | ≤ m ) (cid:27) ordered by q ≤ p iff ∀ i < ω ( p ( i ) ⊆ q ( i )). This poset adds a slalom ˙ ϕ ∗ ∈ Q i<ω [ b ( i )] ≤ h ( i ) ,defined by ˙ ϕ ∗ ( i ) := S p ∈ ˙ G p ( i ), such that x ∈ ∗ ϕ ∗ for every x ∈ Q i<ω b ( i ) in the groundmodel. Note that ˙ ϕ ∗ ( i ) is forced to either have size h ( i ) (whenever h ( i ) ≤ b ( i )) or to beequal to b ( i ) (whenever b ( i ) ≤ h ( i )).This poset is Suslin σ -Fr-linked. In fact, for any s ∈ S n<ω Q i Let ν be a cardinal of uncountable cofinality and let s be a standardcoherent system that satisfies the hypothesis of Lemma 2.7 for θ = ν . Further assumethat ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 17 (i) Γ ⊆ Ω has size ≥ ν ,(ii) D ∈ I and Γ ⊆ D ,(iii) for each l ∈ Γ , P D, adds a real ˙ a l in [ ω ] ℵ such that, for any Z ⊆ D in I ∩ [Ω] <ν , ˙ A (cid:22) Z := h ˙ a l : l ∈ Z ∩ Γ i is a P Z, -name and, whenever l ∈ D r Z , P D, forces that ˙ a l is Md ( ˙ A (cid:22) Z ) -unbounded over V Z, , and(iv) for every ξ ∈ S s and B ∈ I ∩ [ D ] <ν , P B,ξ forces that ˙ Q B,ξ is either uniformly Md ( ˙ A (cid:22) B ) -good or a random algebra.Then P D,π forces that any infinite subset of ω intersects some member of ˙ A := ˙ A (cid:22) Γ . Inparticular, P D,π forces a ≤ | Γ | whenever ˙ A is an a.d. family.Proof. Let ˙ x be a P D,π -name of an infinite subset of ω . By Lemma 2.7, there is some Z ⊆ D in I ∩ [Ω] <ν such that ˙ x is a P Z,π -name. Thus, by Lemmas 3.19, 3.20 andTheorem 3.23, for any l ∈ Γ r Z , P D,π forces that ˙ a l is Md ( ˙ A (cid:22) Z )-unbounded over V Z,π ,so ˙ x / ∈ I ( ˙ A (cid:22) Z ) implies that ˙ x ∩ ˙ a l is infinite by Lemma 3.17. As ˙ x ∈ I ( ˙ A (cid:22) Z ) implies that˙ x ∩ ˙ a j is infinite for some j ∈ Z ∩ Γ, we are done. (cid:3) Remark 3.33. By Theorem 3.27, it is clear that in condition (iv) of Theorem 3.32 wecan use Suslin σ -Fr-linked posets. 4. Applications The following result generalizes [FFMM18, Thm. 4.17] in the sense that it allows topreserve mad families of singular cardinality along a more general type of FS iterations. Theorem 4.1. Let θ be an uncountable regular cardinal and let Ω be a set of size ≥ θ .After forcing with H Ω , any further FS iteration where each iterand is one of the followingtypes preserves the mad family added by H Ω .(0) Suslin σ - Fr -linked.(1) Random algebra.(2) Hechler poset (for adding a mad family).(3) Poset with ccc of size < θ .Proof. Consider a FS iteration P π = h P ξ , ˙ Q ξ : ξ < π i with π > Q = P = H Ω and, for 0 < ξ < π , ˙ Q ξ is of one of the types above. To be more precise, let h C j : j < i be a partition of [1 , π ) such that, for each j < ξ ∈ C j , ˙ Q ξ is a P α -name of a posetof type ( j ). Note that this iteration can be defined as the standard coherent system m on I m := hP (Ω) , ⊆i such that(o) P X, = H X for any X ⊆ Ω;(i) S m = C ∪ C , C m = C ∪ C ;(ii) ∆ : [1 , π ) → [Ω] <θ such that ∆( ξ ) = ∅ for each ξ ∈ C ∪ C ;(iii) for ξ ∈ C , when ˙ Q ξ is coded in V ∆( ξ ) ,ξ , ˙ S ξ = ˙ Q ξ (or trivial otherwise, though thislatter case will not happen) and, for ξ ∈ C , S ξ is the random algebra ˙ Q ξ (wlog itssupport is in the ground model);(iv) for ξ ∈ C , ˙ Q m ξ = ˙ Q ξ (wlog, the support of this Hechler poset is in the ground model)and, for ξ ∈ C , when ˙ Q ξ is forced to be in V ∆( ξ ) ,ξ , ˙ Q m ξ = ˙ Q ξ (or trivial otherwise,though this will not happen).By recursion on ξ ≤ π , m (cid:22) ξ and ∆ (cid:22) ξ should be constructed and it should be guaranteedthat P Ω ,ξ = P ξ . This is fine in the steps ξ = 0 , ξ . As ξ ∈ C j for some j < we consider cases for each j . If j = 0 then, as a Suslin ccc poset is coded by reals and P Ω ,ξ = P ξ , by Lemma 2.7 there is some ∆( ξ ) ∈ [Ω] <θ such that ˙ Q ξ is (coded by) a P ∆( ξ ) ,ξ -name; if j = 1 , Q ξ can be assumed to be in the groundmodel, we can put ∆( ξ ) = 0; if j = 3 then, by Lemma 2.7 and the regularity of θ , there issome ∆( ξ ) ∈ [Ω] <θ such that ˙ Q ξ is a P ∆( ξ ) ,ξ -name. Therefore, in any case, we can define m (cid:22) ( ξ + 1) as required and it is clear that P Ω ,ξ = P ξ .Now let ˙ A = h ˙ a l : l ∈ Ω i be the P Ω , -name of the generic a.d. family it adds. Note that˙ a l is a P { l } , -name and, by Lemma 3.18, for any B ⊆ Ω with l / ∈ B , P B ∪{ l } , forces that ˙ a l is Md ( ˙ A (cid:22) B )-unbounded over V B, . Hence, by Theorem 3.32, P Γ ,π forces that ˙ A is a madfamily. (cid:3) Remark 4.2. The previous theorem remains true if we add the type(0’) Suslin ccc poset coded in the ground model such that, for any X ⊆ Ω in the groundmodel, it is uniformly Md ( ˙ A (cid:22) X )-good in any ccc generic extension of V H X .By Theorem 3.27 Suslin σ -Fr-linked posets coded in the ground model satisfy (0’).The remaining results in this section are improvements of the consistency results of[FFMM18, Sect. 5] about separating cardinals in Cicho´n’s diagram. Not only can weforce an additional singular value, but the constructions are uniform in the sense thatthere is no need to distinguish between 2D or 3D constructions anymore since all thecoherent systems can be constructed on a partial order of the form hP (Ω) , ⊆i . In thefollowing proofs, sum and product denote the corresponding operations in the ordinals,even when they are applied to cardinal numbers.The following result improves [FFMM18, Thm. 5.2] about separating Cicho´n’s diagraminto 7 different values. Theorem 4.3. Assume that θ ≤ θ ≤ κ ≤ µ are uncountable regular cardinals, ν ≤ λ are cardinals such that µ ≤ ν , ν <κ = ν and λ <θ = λ . Then there is a ccc poset thatforces MA <θ , add( N ) = θ , cov( N ) = θ , b = a = κ , non( M ) = cov( M ) = µ , d = ν and non( N ) = c = λ .Proof. Let Ω and Ω be disjoint sets of size κ and ν respectively. Put Ω := Ω ∪ Ω . As ν <κ = ν , we can enumerate [Ω] <κ := { W ζ : ζ < ν } . Fix a bijection g = ( g , g , g ) : λ → × ν × λ and a function t : νµ → ν such that, for any ζ < ν , t − [ { ζ } ] is cofinal in νµ ( ).Put π := λνµ , S := { λρ : ρ < νµ } and define ∆ : [1 , π ) → [Ω] <κ such that ∆( λρ ) := ∅ ,∆( λρ + 1) := W t ( ρ ) and ∆( λρ + 2 + ε ) = W g ( ε ) for each ρ < νµ and ε < λ .Define the standard coherent system m of FS iterations of length π on hP (Ω) , ⊆i suchthat S m := S , C m := [1 , π ) r S , ∆ m := ∆, ˙ Q m X, := H X ∩ Ω × C X ∩ Ω and where theFS iterations at each interval of the form [ λρ, λ ( ρ + 1)) for ρ < µν is defined as follows.Assume that m (cid:22) λρ has already been defined. For each ζ < ν choose(0) an enumeration { ˙ Q ,ζ,γ : γ < λ } of all the (nice) P W ζ ,λρ -names for posets of size < θ ,with underlying set contained in θ , that are forced by P Ω ,λρ to have ccc; and(1) an enumeration { ˙ Q ,ζ,γ : γ < λ } of all the (nice) P W ζ ,λρ -names for subalgebras ofrandom forcing of size < θ .Put S m λρ := E (only when ρ > 0) and, for each ξ ∈ ( λρ, λ ( ρ + 1)), put(i) ˙ Q m ξ := D V ∆( ξ ) ,ξ when ξ = λρ + 1, and(ii) ˙ Q m ξ := ˙ Q g ( ε ) when ξ = λρ + 2 + ε for some ε < λ . For example, define t ( νδ + α ) = α for each δ < µ and α < ν ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 19 This construction is possible because, as λ <θ = λ , each P Ω ,ξ has size ≤ λ .It remains to show that P := P Ω ,π forces what we want. First note that this posetcan be obtained by the FS iteration h P Ω ,ξ , ˙ Q Ω ,ξ : ξ < π i , and observe that all theseiterands are θ - Lc -good and θ - Cn -good. Hence, by Theorem 3.10, P forces add( N ) ≤ θ ,cov( N ) ≤ θ and λ ≤ non( N ). Actually, those are equalities, even more, P forces MA <θ (which implies add( N ) ≥ θ ). To see this, let ˙ R is a P -name of a ccc poset of size < θ and ˙ D a family of size < θ of dense subsets of ˙ R . By Lemma 2.7 there is some ζ < ν suchthat both ˙ R and ˙ D are P W ζ ,π -names. Moreover, as π has cofinality µ , there is some ρ < νµ such both are P W ζ ,λρ -names. Therefore, there is some γ < λ such that ˙ R = ˙ Q ,ζ,γ , so thegeneric set added by ˙ Q g ( ε ) = ˙ Q m W ζ ,ξ intersects all the dense sets in ˙ D where ε := g − (0 , ζ, γ )and ξ := λρ + 2 + ε . In a similar way, it can be proved that P forces cov( N ) ≥ θ . On theother hand, since (cid:13) P c ≤ λ follows from | P | ≤ λ , together with non( N ) ≥ λ (see above)it is forced that non( N ) = c = λ .As the FS iteration that determines P has cofinality µ and µ -cofinally many full eventu-ally different reals are added by E , P forces cov( M ) ≤ µ ≤ non( M ). Actually, non( M ) ≤ µ ≤ cov( M ) is forced by Theorem 3.10 applied to the Prs Ed , so non( M ) = cov( M ) = µ .Now we show that P forces a ≤ κ and ν ≤ d . Let ˙ A := { ˙ a l : l ∈ Ω } be the H Ω -nameof the mad family added by H Ω and let { ˙ c l : l ∈ Ω } ⊆ ω ω be the Cohen reals addedby C Ω . For any X ⊆ Ω, l ∈ Ω and l ∈ Ω , it is clear that ˙ a l is a P X, -name whenever l ∈ X , and ˙ c l is a P X, -name whenever l ∈ X . On the other hand, if l / ∈ X then P X ∪{ l } , forces that ˙ c l is Cohen over V X, , hence it is D -unbounded over it; and if l / ∈ X then P X ∪{ l } , forces that ˙ a l is Md ( ˙ A (cid:22) ( X ∩ Ω ))-unbounded over V X, . The latter is a conse-quence of Lemma 3.20 applied to C X ∩ Ω . Therefore, by Theorems 3.15 and 3.32 appliedto { ˙ c l : l ∈ Ω } and { ˙ a l : l ∈ Ω } respectively, P forces ν ≤ d (because { ˙ c l : l ∈ Ω } isstrongly κ - D -unbounded) and a ≤ κ .It remains to show that P forces κ ≤ b and d ≤ ν . For each ρ < νµ denote by ˙ d ρ the (restricted) dominating real over V W t ( ρ ) ,λρ +1 added by ˙ Q W t ( ρ ) ,λρ +1 . It is enough toshow that P forces that any subset of ω ω of size < κ is dominated by some ˙ d ρ (hence { ˙ d ρ : ρ < νµ } is a dominating family of size ν ). Let ˙ F be a P -name of such a subset of ω ω . By Lemma 2.7 and because cf( π ) = µ , there are ζ < ν and ρ < νµ such that ˙ F isa P W ζ ,λρ -name. Thus, there is some ρ ∈ [ ρ , νµ ) such that t ( ρ ) = ζ , so P W ζ ,λρ +2 forcesthat ˙ d ρ dominates ˙ F . (cid:3) We summarize in the rest of this section the results from [Mej13a] and [FFMM18, Sect.5] that can be improved by the method of the previous proof. Note that, in the forcingconstructions for Theorems 4.5(b) and 4.6(c),(d), we cannot preserve a mad family addedby a poset of the form H Ω because their constructions require that full generic dominatingreals are added. For these items, it is enough to base their constructions on hP ( ν ) , ⊆i andstart with ˙ Q X, := C X for any X ⊆ ν . In addition, by an argument similar to [FFMM18,Rem. 5.9], it can be additionally forced within these items that a = µ . Theorem 4.4. Let θ ≤ θ ≤ κ ≤ ν ≤ λ be as in the statement of Theorem 4.3. Thenthere is a ccc poset forcing MA <θ , add( N ) = θ , cov( N ) = θ , b = a = non( M ) = κ , cov( M ) = d = ν and non( N ) = c = λ .Proof. The construction of the standard coherent system that forces the above is verysimilar to the one in the proof of Theorem 4.3. The only changes are that S m := ∅ and, for each ξ ∈ [ λρ, λ ( ρ + 1)), ˙ Q m ξ := D V ∆( ξ ) ,ξ when ξ = λρ , and ˙ Q m ξ := ˙ Q g ( ε ) when ξ = λρ + 1 + ε for some ε < λ . (cid:3) Theorem 4.5. Assume that θ ≤ κ ≤ µ are uncountable regular cardinals, ν ≤ λ arecardinals such that µ ≤ ν , ν <κ = ν and λ <θ = λ . Then, for each of the statements below,there is a ccc poset forcing it.(a) MA <θ , add( N ) = θ , b = a = κ , cov( I ) = non( I ) = µ for I ∈ {M , N } , d = ν and cof( N ) = c = λ .(b) MA <θ , add( N ) = θ , cov( N ) = κ , add( M ) = cof( M ) = µ , non( N ) = ν and cof( N ) = c = λ .(c) MA <θ , add( N ) = θ , cov( N ) = b = a = κ , non( M ) = cov( M ) = µ , d = non( N ) = ν and cof( N ) = c = λ .(d) MA <θ , add( N ) = θ , cov( N ) = b = a = non( M ) = κ , cov( M ) = d = non( N ) = ν and cof( N ) = c = λ . Theorem 4.6. Assume that κ ≤ µ are uncountable regular cardinals, ν ≤ λ are cardinalssuch that µ ≤ ν , ν <κ = ν and λ ℵ = λ . Then, for each of the statements below, there isa ccc poset forcing it.(a) add( N ) = cov( N ) = b = a = κ , non( M ) = cov( M ) = µ , d = non( N ) = cof( N ) = ν and c = λ .(b) add( N ) = b = a = κ , cov( I ) = non( I ) = µ for I ∈ {M , N } , d = cof( N ) = ν and c = λ .(c) add( N ) = cov( N ) = κ , add( M ) = cof( M ) = µ , non( N ) = cof( N ) = ν and c = λ .(d) add( N ) = κ , cov( N ) = add( M ) = cof( M ) = non( N ) = µ , cof( N ) = ν and c = λ .(e) add( N ) = non( M ) = a = κ , cov( M ) = cof( N ) = ν and c = λ .Moreover, if λ <κ = λ , MA <κ can be forced additionally at each of the items above. Remark 4.7. This method can be used to force values (even singular) to other cardinalinvariants different than those from Cicho´n’s diagram. For instance, the results in [Mej17,Sect. 3] can be adapted to the present approach.5. Bonus track: linkedness properties The notions of σ -linked, σ -centered, σ -Fr-linked, etc., can be put into the followinggeneral framework. Definition 5.1. Say that Γ is a linkedness property (for subsets of posets) if Γ is a class-function with domain the class of posets such that, for any poset P , Γ( P ) ⊆ P ( P ) ( ).We define the following notions for a linkedness property Γ.(1) Γ is basic if [ P ] ≤ ⊆ Γ( P ) for any poset P .(2) Γ is conic if, for any poset P , P ⊆ P and Q ∈ Γ( P ), if P ⊆ { p ∈ P : ∃ q ∈ Q ( q ≤ p ) } and Q = { q ∈ Q : ∃ p ∈ P ( q ≤ p ) } then P ∈ Γ( P ).(3) Γ is a downwards forcing linkedness property if, for any complete embedding ι : P → Q between posets, if P ⊆ P and ι [ P ] ∈ Γ( Q ) then P ∈ Γ( P ).(4) Γ is an upwards forcing linkedness property if, for any complete embedding ι : P → Q between posets, if P ∈ Γ( P ) and ι (cid:22) P is 1-1 then ι [ P ] ∈ Γ( Q ).(5) A forcing linkedness property is a downwards and upwards linkedness forcing property.(6) Γ is appropriate if it is a basic conic forcing linkedness property. Concretely, Γ is a formula ϕ ( x, y ) (with fixed parameters) in the language of ZF and Γ( P ) := { Q ⊆ P : ϕ ( Q, P ) } . ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 21 (7) Γ is closed if, for any poset P and Q ⊆ Q ⊆ P , if Q ∈ Γ( P ) then Q ∈ Γ( P ).(8) Let µ be an infinite cardinal. A poset P is µ - Γ -covered if it can be covered by ≤ µ -many sets from Γ( P ). When µ = ℵ we just say σ - Γ -covered. (9) Let θ be an infinite cardinal. A poset P is θ - Γ -Knaster if, for any P ⊆ P of size θ ,there is some Q ⊆ P of size θ such that Q ∈ Γ( P ). For θ = ℵ we just say Γ -Knaster .(10) If Λ is another linkedness property, say that Λ is stronger than Γ (or Γ is weaker than Λ), denoted by Λ ⇒ Γ, if Λ( P ) ⊆ Γ( P ) for any poset P . We say that both propertiesare equivalent , denote by Λ ⇔ Γ, when one is weaker and stronger than the other. Remark 5.2. Let Γ be a linkedness property and µ an infinite cardinal.(1) If Γ is closed, then any µ -Γ-covered poset is µ + -Γ-Knaster.(2) If Γ is a closed conic forcing linkedness property then ‘ µ -Γ-covered’ is a property offorcing notions.(3) If Γ is appropriate and µ is regular, then ‘ µ -Γ-Knaster’ is a property of forcing notions.(4) If a property Λ is stronger than Γ then any µ -Λ-covered (Knaster) poset is µ -Γ-covered(Knaster).To see that closed is necessary in (1) and (2), consider the property Γ defined by Q ∈ Γ ( P ) iff Q is not an antichain in P of size ≥ 2. Note that Γ is an appropriate linkednessproperty, but it is not closed. A poset P is µ -Γ -covered iff either it is an antichain initself of size ≤ µ , or it is not an antichain in itself. Hence, though ω ≤ and ω (as posetsof sequences in ω ordered by ⊇ ) are forcing equivalent, the first is σ -Γ -covered while thesecond is not. On the other hand, any poset is µ -Γ -Knaster iff it is µ -cc. Example 5.3. The following are appropriate closed linkedness properties.Γ η -cc (when 2 ≤ η ≤ ω ): η -cc, that is, Q ∈ Γ η -cc ( P ) iff Q does not contain antichains in P of size η .Γ bd-cc : n -cc for some 2 ≤ n < ω .Λ n (when 2 ≤ n < ω ): n -linked.Λ ω : centered.Γ cone : Say that Q ∈ Γ cone ( P ) if there is some q ∈ P such that ∀ p ∈ Q ( q ≤ ∗ p ) ( ).Λ Fr : Frechet-linked.It is clear that Γ cone ⇒ Λ ω ⇒ Λ n +1 ⇒ Λ n ⇒ Λ ⇒ Γ n -cc ⇒ Γ n +1-cc ⇒ Γ bd-cc ⇒ Γ ω -cc for 2 ≤ n < ω (actually Γ ⇔ Λ ). Also Γ cone ⇒ Λ Fr ⇒ Γ ω -cc and Λ ⇒ Γ . Theseproperties determine some well-known forcing properties, for example, ‘ µ -Λ ω -covered’means ‘ µ -centered’, ‘ µ -Λ ω -Knaster’ means ‘precaliber µ ’, ‘ µ -Λ -covered’ means ‘ µ -linked’,‘ µ -Λ -Kanster’ is the typical µ -Knaster property, and ‘ µ -Λ Fr -covered’ is what we definedas µ -Fr-linked in Definition 3.24.By an argument similar to [HT48, Thm. 2.4] it can be proved that, if θ is regular,then any Q ∈ Γ ω -cc ( P ) of size θ contains a 2-linked subset of the same size, thus θ -Λ -Knaster is equivalent to θ -Γ ω -cc -Knaster. On the other hand, Todorˇcevi´c [Tod91, Tod86]constructed a Λ ω -Knaster (i.e. ℵ -precaliber) poset that is not σ -Γ ω -cc -covered (i.e. σ -finite-cc) and, under b = ℵ , a σ -Λ n -covered poset that is not Λ n +1 -Knaster. Todorˇcevi´c[Tod14] and Th¨ummel [Th¨u14] constructed σ -finite-cc posets that are not σ -Γ bd-cc -covered(i.e. σ -bounded-cc).It is clear that any Boolean algebra that admits a strictly positive fam (finitely additivemeasure) is σ -bounded-cc and, by Lemma 3.29, any complete Boolean algebra that admits Here, ≤ ∗ denotes the separable order of P , that is, q ≤ ∗ p iff any condition compatible with q in P iscompatible with p . a strictly positive σ -additive measure is σ -Λ Fr -covered. Note that D is a σ -centered posetthat admits a strictly positive fam but it is not σ -Frechet-linked (otherwise it wouldcontradict Theorem 3.30), and B c + is a complete Boolean algebra that admits a strictlypositive σ -additive measure but it is not σ -linked (see Dow and Steprans [DS94]).Note that any poset is µ -Γ cone -covered iff it is forcing equivalent to a poset of size ≤ µ ,and the notion θ -Γ cone -Knaster is equivalent to ( θ, θ )-caliber. The property Γ cone is thestrongest of all the appropriate linkedness properties with respect to the class of separativeposets, so it is morally the strongest appropriate linkedness property.With the exception of Λ Fr , all the other properties (including Γ ) are absolute fortransitive models of ZFC. Recall from [Paw92] that there is a ω ω -bounding proper posetthat forces that B V (random forcing from the ground model) adds a dominating real, sothis poset, though σ -Frechet-linked in the ground model, is not forced to be so. Example 5.4. In the work in progress [BCM] we discuss properties stronger than Λ Fr .Given a free filter F of subsets of ω , define Λ F such that, for any poset P , Q ∈ Λ F ( P )iff for any sequence h p n : n < ω i in Q there is some q ∈ P that forces { n < ω : p n ∈ ˙ G } ∈ F + (that is, it intersects every member of F ), which is an appropriate closedlinkedness property. Also define Λ uf ( P ) := T { Λ F ( P ) : F free filter } . It is clear thatΛ uf ⇒ Λ F ⇒ Λ F ⇒ Λ Fr whenever F ⊆ F . Even more, we have the following equiva-lences. Lemma 5.5. (a) For any free filter F in ω generated by < p -many sets, Λ F ⇔ Λ Fr .(b) For any p -cc poset P , Λ uf ( P ) = Λ Fr ( P ) .Proof. Both items can be proved simultaneously. Let P a poset, F a free filter on ω andassume that either F is generated by < p -many sets or P is p -cc. It is enough to showthat Λ Fr ( P ) ⊆ Λ F ( P ). Assume that Q ⊆ P is Fr-linked but not in Λ F ( P ), so there area countable sequence h p n : n < ω i in Q , a maximal antichain A ⊆ P and a sequence h a r : r ∈ A i in F such that each r ∈ A is incompatible with p n for every n ∈ a r . In any ofthe two cases of the hypothesis, it can be concluded that there is some pseudo-intersection a ∈ [ ω ] ℵ of h a r : r ∈ A i . Hence each r ∈ A forces p n ∈ ˙ G for only finitely many n ∈ a ,which means that P forces the same. However, as Q is Fr-linked, there is some q ∈ P that forces ∃ ∞ n ∈ a ( p n ∈ ˙ G ), a contradiction. (cid:3) As a consequence of the previous result and Lemma 3.29, E and any complete Booleanalgebra that admits a strictly positive σ -additive measure are σ -Λ uf -covered. In [BCM]we show that Λ Fr -Knaster posets do not add dominating reals. Hence, D becomes anexample of a σ -centered poset that is not Λ Fr -Knaster. Also note that these propertiesassociated with filters are not absolute.To finish, in the general context of Definition 5.1, we provide simple conditions tounderstand when the FS iteration of θ -Γ-Knaster (or covered) posets is θ -Γ-Knaster (orcovered), likewise for FS products. These conditions are summarized in the followingdefinition, and they just represent facts extracted from the typical proofs of the iterationresults for σ -linked and Λ -Knaster. At the end of this section, we relate the linkednessproperties presented so far with the notions below. Definition 5.6. Let Γ be a linkedness property.(1) Γ is productive if, for any posets P and Q , and for any Q ⊆ P × Q , if dom Q ∈ Γ( P )and ran Q ∈ Γ( Q ) then Q ∈ Γ( P × Q ). ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 23 (2) Γ is FS-productive if for any sequence h P i : i ∈ I i of posets, n < ω and any Q ⊆ { p ∈ Q <ωi ∈ I P i : | dom p | ≤ n } (FS product), if { dom p : p ∈ Q } forms a ∆-system with root s and { p (cid:22) s : p ∈ Q } ∈ Γ( Q i ∈ s P i ) then Q ∈ Γ (cid:0) Q <ωi ∈ I P i (cid:1) .(3) Γ is strongly productive if, for any sequence h P i : i ∈ I i of posets, Q ∈ Γ (cid:0) Q <ωi ∈ I P i (cid:1) whenever(i) there is some n < ω such that Q ⊆ { p ∈ Q <ωi ∈ I P i : | dom p | ≤ n } and(ii) { p ( i ) : p ∈ Q } ∈ Γ( P i ) for any i ∈ I .(4) Γ is two-step iterative if, for any poset P , any P -name ˙ Q of a poset and any Q ⊆ P ∗ ˙ Q ,if dom Q ∈ Γ( P ) and P forces that { ˙ q : ∃ p ∈ ˙ G (( p, ˙ q ) ∈ Q ) } ∈ Γ( ˙ Q ), then Q ∈ Γ( P ∗ ˙ Q ).(5) Γ is direct-limit iterative if whenever(i) θ is an uncountable regular cardinal,(ii) h P α : α ≤ δ i is an increasing l -sequence of posets such that cf( δ ) = θ and P γ = limdir α<γ P α for any limit γ ≤ δ ,(iii) f : θ → δ is increasing,(iv) Q = { p ξ : ξ < θ } ⊆ P δ such that each p ξ ∈ P f ( ξ +1) , and(v) for each ξ < θ , r ξ ∈ P f ( ξ ) is a reduction of p ξ ,if there is some γ < δ such that r ξ ∈ P γ for every ξ < θ and { r ξ : ξ < θ } ∈ Γ( P γ ),then Q ∈ Γ( P δ ).(6) Γ is strongly iterative if, for any FS iteration P δ = h P α , ˙ Q α : α < δ i , Q ∈ Γ( P δ )whenever(i) there is some n < ω such that Q ⊆ { p ∈ P δ : | dom p | ≤ n } and(ii) for any α < δ , if Q (cid:22) α ∈ Γ( P α ) then (cid:13) P α { p ( α ) : p (cid:22) ( α + 1) ∈ Q (cid:22) ( α + 1) , p (cid:22) α ∈ ˙ G α } ∈ Γ( ˙ Q α ) . Note that any strongly productive linkedness property is both productive and FS-productive. On the other hand, if Γ is strongly iterative, P δ = h P α , ˙ Q α : α < δ i is a FSiteration and Q ⊆ { p ∈ P δ : | dom p | ≤ n } satisfies (6)(i),(ii) then Q (cid:22) α ∈ Γ( P α ) for any α ≤ δ . It is clear that any strongly iterative property is two-step iterative and satisfies aweak form of direct-limit iterative (which we leave implicit in the proof of Corollary 5.10).The following is a general result about FS products. Theorem 5.7. Let µ be an infinite cardinal, θ an uncountable regular cardinal, and let Γ be an appropriate linkedness property.(a) If Γ is productive then any finite product of µ - Γ -covered sets is µ - Γ -covered.(b) If Γ is closed and productive then and any finite product of θ - Γ -Knaster posets is θ - Γ -Knaster.(c) If Γ is FS-productive and h P i : i ∈ I i is a sequence of θ - Γ -Knaster posets, then Q <ωi ∈ I P i is θ - Γ -Knaster iff Q i ∈ s P i is θ - Γ -Knaster for every s ∈ [ I ] < ℵ .(d) If Γ is strongly productive, h P i : i ∈ I i is a sequence of µ - Γ -covered posets and | I | ≤ µ ,then Q <ωi ∈ I P i is µ - Γ -covered.Proof. Items (a),(b) are easy and (c) follows by a classical ∆-system argument. Item (d)uses the following result. Lemma 5.8 (Engelking and Kar lowicz [EK65]) . If µ is an infinite cardinal and I is aset of size ≤ µ then there exists a set H ⊆ µ I of size ≤ µ such that any finite partialfunction from I to µ is extended by some member of H . For each i ∈ I choose a sequence h Q i,ζ : ζ < µ i of non-empty sets in Γ( P i ) that covers P i . Let H be as in Lemma 5.8. By Definition 5.6(3), the set Q ∗ h,n := { p ∈ Q <ωi ∈ I Q i,h ( i ) : | dom p | = n } is in Γ (cid:0) Q <ωi ∈ I P i (cid:1) and it is clear that h Q ∗ h,n : h ∈ H, n < ω i covers Q <ωi ∈ I P i . (cid:3) Now we turn to a general result about FS iterations. Theorem 5.9. Let µ be an infinite cardinal, θ an uncountable regular cardinal, and let Γ be an appropriate linkedness property.(a) If Γ is two-step iterative, P is µ - Γ -covered, and P forces that Γ is basic and that ˙ Q isa | µ | - Γ -covered poset, then P ∗ ˙ Q is µ - Γ -covered.(b) Let P be a θ - Γ -Knaster poset and let ˙ Q be a P -name of a θ - Γ -Knaster poset. Assumein addition that either Γ = Γ , or Γ is closed, two-step iterative, P is θ -cc and P forces that Γ is closed and basic. Then P ∗ ˙ Q is θ - Γ -Knaster.(c) If Γ is direct-limit iterative, δ is a limit ordinal and h P α : α ≤ δ i is an increasing l -sequence of θ - Γ -Knaster posets such that P γ = limdir α<γ P α for any limit γ ≤ δ ,then P δ is θ - Γ -Knaster.(d) If Γ is strongly iterative and any poset forces that Γ is still basic, then any FS iterationof length < (2 µ ) + of µ - Γ -covered posets is µ - Γ -covered.Proof. To see (a), it is enough to show that, for any poset P that forces Γ to be still basicand any P -name ˙ Q for a poset, if P ∈ Γ( P ) and ˙ Q is a P -name of a non-empty set inΓ( ˙ Q ), then P ∗ ˙ Q := { ( p, ˙ q ) ∈ P ∗ ˙ Q : p ∈ P and p (cid:13) ˙ q ∈ ˙ Q } is in Γ( P ∗ ˙ Q ). Clearlydom( P ∗ ˙ Q ) = P . On the other hand, any p ∈ P forces that ˙ R := { ˙ q : ∃ p ∈ ˙ G (( p, ˙ q ) ∈ P ∗ ˙ Q ) } = ˙ Q , so P forces that ˙ R is either ˙ Q or the empty set, so ˙ R ∈ Γ( ˙ Q ). By Definition5.6(4), it follows that P ∗ ˙ Q ∈ Γ( P ∗ ˙ Q ).Item (b) is well-known when Γ = Γ , so assume that Γ is closed, two-step iterative, P is θ -cc and P forces that Γ is closed and basic. Let { ( p α , ˙ q α ) : α < θ } ⊆ P ∗ ˙ Q . As P forces ˙ Q to be θ -Γ-Knaster, there is some P -name ˙ K for a subset of θ such that P forces that “ whenever |{ α < θ : p α ∈ ˙ G }| = θ , ˙ K ⊆ { α < θ : p α ∈ ˙ G } has size θ and { ˙ q α : α ∈ ˙ K } ∈ Γ( ˙ Q ), otherwise ˙ K = ∅ ”. Set K := { α < θ : α / ∈ ˙ K } , which has size θ (otherwise P would force that |{ α < θ : p α ∈ ˙ G }| < θ , which contradicts that P is θ -cc).For each α ∈ K choose an r α ≤ p α that forces α ∈ ˙ K . Hence, there is some K ⊆ K ofsize θ such that { r α : α ∈ K } ∈ Γ( P ).As Γ is conic and r α ≤ p α for any α ∈ K , it is enough to show that Q := { ( r α , ˙ q α ) : α ∈ K } ∈ Γ( P ∗ ˙ Q ). It is clear that dom Q ∈ Γ( P ). On the other hand, P forces that˙ R := { ˙ q α : r α ∈ ˙ G, α ∈ K } ⊆ { ˙ q α : α ∈ ˙ K } , so ˙ R ∈ Γ( ˙ Q ) because Γ is closed. As Γ istwo-step iterative, we are done.Now we show (c). Let { p ξ : ξ < θ } ⊆ P δ . If cf( δ ) = θ then there are some α < δ anda K ⊆ θ of size θ such that { p ξ : ξ ∈ K } ⊆ P α , so there is some K ⊆ K of size θ suchthat { p ξ : ξ ∈ K } ∈ Γ( P α ) (note that Γ( P α ) ⊆ Γ( P δ ) because Γ is appropriate). Assumethat cf( δ ) = θ and choose an increasing continuous cofinal function g : θ → δ such thateach g ( ξ ) is a limit ordinal. For each ξ < θ choose a reduction r ξ ∈ P g ( ξ ) of p ξ . As g ( ξ )is limit, there is some h ( ξ ) < ξ such that r ξ ∈ P g ( h ( ξ )) . Hence, by Fodor’s Lemma, thereis some stationary set S ⊆ θ such that h [ S ] = { η } for some η < θ , that is, r ξ ∈ P g ( η ) forevery ξ ∈ S . By recursion define j : θ → S increasing such that j (0) > η and, for any ζ < θ , p j ( ζ ) ∈ P g ( j ( ζ +1)) . As P g ( η ) is θ -Γ-Knaster, there is some K ⊆ θ of size θ such that { r j ( ζ ) : ζ ∈ K } ∈ Γ( P g ( η ) ). Let i : θ → K be the increasing enumeration of K . ATRIX ITERATIONS WITH VERTICAL SUPPORT RESTRICTIONS 25 Put f := g ◦ j ◦ i and γ := g ( η ). Note that { p j ( i ( β )) : β < θ } , { r j ( i ( β )) : β < θ } , f and γ satisfy the conditions of Definition 5.6(5) so, as Γ is direct-limit iterative, { p j ( i ( β )) : β <θ } ∈ Γ( P δ ).To finish, we show (d). Let δ < (2 µ ) + and let P δ = h P α , ˙ Q α : α < δ i be a FS iterationof µ -Γ-covered sets. For each α < δ choose a sequence h ˙ Q α,ζ : ζ < µ i of P α -names ofsets in Γ( ˙ Q α ) that is forced to cover ˙ Q α . For α ≤ δ define P ∗ α ⊆ P α such that p ∈ P ∗ α iff p ∈ P α and, for any ξ ∈ dom p , there is some ζ < µ such that p (cid:22) ξ (cid:13) P ξ p ( ξ ) ∈ ˙ Q ξ,ζ . Byinduction it can be proved that P ∗ α is dense in P α .Now choose H as in Lemma 5.8 and, for each h ∈ H and n < ω , define Q h,n as theset of p ∈ P ∗ δ such that | dom p | ≤ n and, for any α ∈ dom p , p (cid:22) α (cid:13) P α p ( α ) ∈ ˙ Q α,h ( α ) . Itis clear that h Q h,n : h ∈ H, n < ω i covers P ∗ δ , so it remains to show that Q h,n ∈ Γ( P δ ).If α < δ and Q h,n (cid:22) α ∈ Γ( P α ) then a similar argument as in (a) shows that P α forces { p ( α ) : p (cid:22) ( α + 1) ∈ Q h,n (cid:22) ( α + 1) , p (cid:22) α ∈ ˙ G } ∈ Γ( ˙ Q α ). Therefore, as Γ is strongly iterative, Q h,n ∈ Γ( P δ ). (cid:3) Corollary 5.10. Let θ an uncountable regular cardinal and assume that Γ is either(i) Γ or(ii) a closed appropriate linkedness property that is closed and basic in any generic ex-tension, and that it is either strongly iterative, or two-step and direct-limit iterative.Then any FS iteration of θ - Γ -Knaster θ -cc posets is θ - Γ -Knaster.Proof. Case (i) and case (ii) when Γ is two-step and direct-limit iterative follow directlyfrom Theorem 5.9. Case (ii) when Γ is strongly iterative is a bit similar but requires abit more work. If h P α , ˙ Q α : α < δ i is a FS iteration of θ -Γ-Knaster θ -cc posets, it isenough to show by induction on α ≤ δ that, for any sequence h p β : β < θ i in P α thereare some K ⊆ θ of size θ and some sequence h r β : β ∈ K i in P α that satisfies (i) and(ii) of Definition 5.6(6) (with respect to P α ) and such that r β ≤ p β for any β ∈ K . Thesuccessor step is exactly like the proof of Theorem 5.9(b) and the limit step is very similarto Theorem 5.9(c). We just look at the case cf( α ) = θ . Let h p β : β < θ i be a sequence in P α . Exactly like in the proof of Theorem 5.9(c), we can find a γ < α , a K ⊆ θ of size θ and an increasing function f : K → α r γ such that, for each β ∈ K , p β (cid:22) f ( β ) ∈ P γ and p β ∈ P f ( β +1) . Even more, we may assume that there is some n < ω such that | dom p β r f ( β ) | = n for all β ∈ K . By the inductive hypothesis, there are K ⊆ K of size θ and a sequence h r β : β ∈ K i of conditions in P γ that satisfies (i) (for some n < ω ) and(ii) of Definition 5.6(6) (with respect to P γ ) and such that r β ≤ p β (cid:22) f ( β ) for any β ∈ K .The set { r β ∪ p β (cid:22) ( f ( β + 1) r f ( β )) : β ∈ K } is as required. (cid:3) Remark 5.11. Table 1 illustrates which productive or iterative notions are satisfied bythe linkedness properties discussed so far.We explain some of the facts indicated in the table. First, we show that Γ ω -cc isstrongly productive. Let Q ⊆ { p ∈ Q <ωi ∈ I P i : | dom p | ≤ n } and assume that { p ( i ) : p ∈ Q } ∈ Γ ω -cc ( P i ) for every i ∈ I . Fix a countable sequence h p k : k < ω i in Q . As the size ofthe domains of the members of the sequence are bounded by n , we can find a W ∈ [ ω ] ℵ such that h p k : k ∈ W i forms a ∆-system with root R . By Ramsey’s Theorem it can beproved that { p k (cid:22) R : k ∈ W } is not an antichain, so neither is h p k : k < ω i .Ramsey’s Theorem also implies that Γ bd-cc is productive. However, it is not FS-productive (consider the set of conditions with domain of size 1 of the FS product Q <ωi ∈ ω P i where each P i is an antichain of size i + 1). The following example indicates that bothΓ ω -cc and Γ bd-cc are not two-step iterative. Consider P := { ( p n , ˙ q n ) : n < ω } ⊆ C ∗ ˙ C such Γ Γ n -cc Γ ω -cc Γ bd-cc Λ n Λ ω Γ cone Λ Fr Λ F Λ uf (3 ≤ n ) (2 ≤ n )Prod. × × (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) ? ? (cid:13) FS Prod. (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) × (cid:13) (cid:13) (cid:13) Str. Prod. × × (cid:13) × (cid:13) (cid:13) × ? ? ?Two-step it. (cid:13) × × × (cid:13) (cid:13) (cid:13) (cid:13) ? (cid:13) Dir.-lim. it. (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) × (cid:13) (cid:13) (cid:13) Str. it (cid:13) × × × (cid:13) (cid:13) × (cid:13) ? ? Table 1. A circle means that the linkedness property satisfies the corre-sponding productive or iterative notion (see Definition 5.6) on the left, an × means that such notion is not satisfied, and a question mark means un-clear.that each p n is a sequence of zeros of length n + 1, p n (cid:13) ˙ q n = h n i but r (cid:13) ˙ q n = h i for any r ∈ C incompatible with p n . Though P is an antichain in C ∗ ˙ C , dom P is centered and C forces that { ˙ q n : p n ∈ ˙ G, n < ω } is a finite antichain.As C ω is uncountable and it has not ( ℵ , ℵ )-precaliber, Theorems 5.7(c),(d) and5.9(c),(d) cannot be applied to Γ cone . Hence, Γ cone does not satisfy the properties in-dicated with × in the table.It is unclear whether Λ F is productive in general, but it is proved in [BCM] that it iswhen F is an ultrafilter. Therefore, Λ uf is productive. By a ∆-system argument, Λ Fr isstrongly iterative. To see this, assume that P δ , Q and n < ω satisfy the conditions inDefinition 5.6(6). It is enough to show, by induction on α ≤ δ , that Q (cid:22) α ∈ Λ Fr ( P α ).Since Λ Fr is two-step iterative, we only need to prove the limit step. Let h p k : k < ω i be a sequence in Q (cid:22) α . As | dom p k | ≤ n for any k < ω , there is some infinite W ⊆ ω such that { dom p k : k ∈ W } forms a ∆-system with root R , and there is some ξ < α such that R ⊆ ξ . By the inductive hypothesis, Q (cid:22) ξ ∈ Λ Fr ( P ξ ), so there is some q ∈ P ξ that forces ∃ ∞ k ∈ W ( p k (cid:22) ξ ∈ ˙ G ξ ). Therefore, it can be proved that q forces (in P α ) that ∃ ∞ k ∈ W ( p k ∈ ˙ G α ).By a similar argument, if Λ Fr were productive then it would be strongly productive.In particular, by Lemma 5.5, Λ Fr restricted to the class of Knaster posets is stronglyproductive, so Theorem 5.7 is valid for Λ Fr for FS products of Knaster posets (or just FSproducts that have ccc). Acknowledgments. This work was supported by grant no. IN201711, Direcci´on Oper-ativa de Investigaci´on, Instituci´on Universitaria Pascual Bravo, and by the Grant-in-Aidfor Early Career Scientists 18K13448, Japan Society for the Promotion of Science.The author would like to thank Miguel Cardona for the very useful discussions thathelped this work to take its final form. He is also very grateful with the anonymous refereefor his/her very useful comments, specially for asking whether any σ -Fr-linked poset is Lc -good (which is answered negative with a counterexample in Remark 3.31). 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