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On cardinal characteristics of Yorioka ideals

Yorioka [J. Symbolic Logic 67(4):1373-1384, 2002] introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in ZFC. We construct a matrix iteration of ccc posets to force that, for many ideals in that class, their associated cardinal invariants (i.e. additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the additivity and cofinality (respectively) of the ideal of Lebesgue measure zero subsets of the real line.


Introduction
Yorioka [Yor02] introduced a characterization of SN , the σ-ideal of strong measure zero subsets of the Cantor space 2 ω , in terms of σ-ideals I f parametrized by increasing functions f ∈ ω ω , which we call Yorioka ideals (see Definition 2.13). Concretely, SN = {I f : f ∈ ω ω increasing} and I f ⊆ N where N is the σ-ideal of Lebesgue-measure zero subsets of 2 ω . Yorioka used this characterization to show that no inequality between cof(SN ) and c := 2 ℵ 0 cannot be decided in ZFC, even more, he proved that cof(SN ) = d κ (the dominating number on κ κ ) whenever add(I f ) = cof(I f ) = κ for all increasing f ( 1 ).
Further research on Yorioka ideals has been continued by Kamo and Osuga [Osu06,KO08,Osu08,KO14]. In [KO08] they proved that, in ZFC, add(I f ) ≤ b ≤ d ≤ cof(I f ) for all increasing f and that, for any fixed f , the basic diagram of the cardinal invariants associated to I f (see Figure 1) is complete in the sense that no other inequality can be proved in ZFC. On the other hand, in [KO14] they constructed models by FS (finite support) iterations of ccc posets where infinitely many cardinal invariants of the form cov(I f ) are pairwise different. Moreover, if there exists a weakly inaccessible cardinal, then there is a ccc poset forcing that there are continuum many pairwise different cardinals of the form cov(I f ).
To continue this line of research, we aim to obtain further consistency results considering several cardinal invariants associated with Yorioka ideals at the same time, that is, to construct models of ZFC where three or more of such cardinal invariants are pairwise different. Given a family I of subsets of a set X, the cardinal invariants associated with I are the four cardinals add(I), cov(I), non(I) and cof(I). The main objective of this paper is to prove the following result. Theorem A. There is a function f 0 ∈ ω ω and a ccc poset forcing that the four cardinal invariants associated with I f are pairwise different for each increasing f ≥ * f 0 .
Concerning problems of this nature, the consistency of add(N ) < cov(N ) < non(N ) < cof(N ) with ZFC is a consequence of [Mej13,Thm. 17]. Quite recently, Goldstern, Kellner and Shelah [GKS] showed the consistency, modulo strongly compact cardinals, of Cichoń's diagram separated into 10 different values, in particular, the four cardinal invariants associated with the meager ideal M on R are pairwise different is consistent. However, this consistency result alone is unknown without using large cardinals.
Though many inequalities between the cardinal invariants associated with Yorioka ideals and the cardinals in Cichoń's diagram are known in ZFC, there are still many open questions. Most of these inequalities had been settled in [KO08,KO14], however, there are a couple of statements from [Osu08] whose proofs do not appear anywhere. Namely, add(N ) ≤ add(I g ) ≤ add(I f ) and cof(I f ) ≤ cof(I g ) ≤ cof(N ) when f, g ∈ ω ω are increasing and f (n + 1) − f (n) ≤ g(n + 1) − g(n) for all but finitely many n < ω. We offer our own proofs of these inequalities in Corollaries 3.14 and 3.16, even more, in Theorem 3.13 we show the stronger fact ( 2 ) that add(I g ) is above some definable cardinal invariant above add(N ) (and dually for cof(I f )). We use this to prove the following new consistency result.
Theorem B (Theorem 5.12). If f ∈ ω ω is increasing then there is a ccc poset that forces add(N ) < add(I f ) < cof(I f ) < cof(N ).
This definable cardinal we use to prove the theorem above is a cardinal characteristic, denoted by b Lc b,h , that is parametrized by functions b, h ∈ ω ω , which form part of what we call localization cardinals (Definition 2.3). These cardinals are a generalization of the cardinal invariants Bartoszyński used to characterize add(N ) and cof(N ) in terms of a slalom structure (Theorem 2.4), and they were used by Brendle and the second author in [BM14] to show consistency results about cardinal invariants associated with Rothberger gaps in F σ ideals on ω. In one of these results, it was constructed a ccc poset that forces infinitely many cardinals of the form b Lc b,h to be pairwise different (even continuum many modulo a weakly inaccessible). An older precedent is known for d Lc b,h , the dual of b Lc b,h : Goldstern and Shelah [GS93] constructed a proper poset (using creatures) to force that ℵ 1 -many cardinals of the form d Lc b,h are pairwise different, result that was later improved by Kellner [KS09] for continuum many. In this work, to prove Theorem B, we show in Theorem 3.13 a connection between the cardinals of type add(I f ), cof(I f ) and the localization cardinals.
A variation of the localization cardinals that we call anti-localization cardinals, denoted by b aLc b,h and (its dual) d aLc b,h (for b, h ∈ ω ω ), also play an important role in the study of cardinal invariants associated with Yorioka ideals. The cardinal d aLc b,h is known from Miller's [Mil81] characterization non(SN ) = min{d aLc b,h : b ∈ ω ω } (for any h ≥ * 1). On the other hand, Kellner and Shelah [KS12] constructed a proper poset that forces that, for the types b aLc b,h and d Lc b,h , continuum many cardinals of each type are pairwise different. In fact, Kamo and Osuga [KO14] discovered a connection between the cardinals of type cov(I f ), non(I f ) and the anti-localization cardinals (Lemmas 3.19 and 3.20), which they use to construct a finite support iteration of ccc posets that forces that, for the types cof(I f ) and b aLc b,1 , infinitely many cardinals of each type are pairwise different (even continuum many when a weakly inaccessible cardinal is assumed).
We use the connections between the cardinal invariants associated with Yorioka ideals and the localization and anti-localization cardinals to prove Theorems A and B. Even more, in the consistency result of the first theorem we can additionally include that infinitely many cardinals of each type b Lc b,h and b aLc b,1 are pairwise different. Theorem C (Theorem 5.7). There is a function f 0 ∈ ω ω and a ccc poset forcing that (a) the four cardinal invariants associated with I f are pairwise different for each increasing f ≥ * f 0 , (b) infinitely many cardinals of the form b Lc b,h are pairwise different, and (c) infinitely many cardinals of the form b aLc b,h and cov(I f ) are pairwise different. Within this result, we merge with Theorem A the consistency result of infinitely many localization cardinals from [BM14], and the consistency result of infinitely many antilocalization cardinals and cov(I f ) from [KO14]. This result is proved by using a FS iteration, so it forces that non(M) ≤ cov(M). Because of this, as ZFC proves that b aLc b,h ≤ non(M) and cov(M) ≤ non(SN ) ≤ non(I f ) for any b, h, f , we cannot expect continuum many values in (b) and (c) above. However, Theorem 5.7 is stated in such generality that (c) can be obtained for continuum many values when a weakly inaccessible is assumed (of course, in this case (a) can not forced). The same applies for (b) but not simultaneously with (c) because of the limitation in our result that the values for the cardinals in (b) appear below those of (c). A curious fact is that, in our result, any single value from (b) and (c) can be repeated continuum many times for different parameters.
The forcing method we use to prove Theorems A, B and C is the technique of matrix iterations to construct two dimensional arrays of posets by FS iterations. This method has been very useful to obtain models where several cardinal characteristics of the continuum are pairwise different. It was introduced for the first time by Blass and Shelah [BS89] to prove the consistency of u < d with large continuum (i.e. c > ℵ 2 ) where u is the ultrafilter number. Later on, this method was improved by Brendle and Fischer [BF11] when they proved the consistency of b = a < s and ℵ 1 < s = b < a with large continuum, the latter assuming the existence of a measurable cardinal in the ground model. The second author [Mej13] induced known preservation results in such type of iterations to construct models where several cardinals in Cichoń's diagram are pairwise different. Just a while ago, Dow and Shelah [DS18] used this technique to prove that the splitting number s is consistently singular.
To guarantee that the matrix iteration constructed to prove the theorem forces the desired values for the cardinal invariants, we propose a more general version of the classical preservation theory of Judah and Shelah [JS90] and Brendle [Bre91], along with the corresponding version for matrix iterations of the second author [Mej13]. This generalization also looks to describe the preservation property that Kamo and Osuga [KO14] proposed to control the covering of Yorioka ideals. Concretely, they proposed a preservation property for cardinal invariants of the form b aLc b,h idω in order to decide values of cov(I f ) thanks to the relations they discovered between both types of cardinals. Although this preservation property is similar to Judah-Shelah's and Brendle's preservation theory, it is not a particular case of it. In view of this, we generalize this classical preservation theory so that the preservation property of Kamo and Osuga becomes a particular case (Example 4.20). Even more, our theory also covers the preservation property defined in [BM14,Sect. 5] to decide cardinal invariants related to Rothberger gaps by forcing (Example 4.26). In addition, we add a particular case of our theory to deal directly with the preservation of cardinals of the form b Lc b,h (Example 4.21). This paper is structured as follows. In Section 2 we review the essential notions of this paper: localization and anti-localization cardinals, Yorioka ideals and related forcing notions and their properties. In Section 3 we show the connection between add(N ), cof(N ) and the localization cardinals, and we review some inequalities between the cardinal invariants associated to Yorioka ideals that are known in ZFC. Additionally, we show what happens to the localization and anti-localization cardinals in non-standard cases, that is, in cases where the function b is allowed to take uncountable values. Section 4 is devoted to our general preservation theory. In Section 5 we prove the main results of this paper. Finally, we present in Section 6 discussions and open questions related to this work.

Notation and preliminaries
Throughout this text, we refer to members of any uncountable Polish space as reals. We write ∃ ∞ n < ω and ∀ ∞ n < ω to abbreviate 'for infinitely many natural numbers' and 'for all but finitely many natural numbers', respectively. For sets A, B, denote by B A the set of functions from A to B. For functions f, g from ω into the ordinals, f ≤ g means that f (n) ≤ g(n) for all n < ω. We say that g (eventually) dominates f , denoted by ). Define f < g and f < * g similarly. F ⊆ ω ω is bounded if it is dominated by a single function in ω ω , i.e, there is a g ∈ ω ω such that f ≤ * g for all f ∈ F . A not bounded set is called unbounded. A set D ⊆ ω ω is dominating if every f ∈ ω ω is dominated by some member of D.
We extend some known operations in the natural numbers as operations between functions of natural numbers defined point-wise. For example, for f, g ∈ ω ω , f + g ∈ ω ω is defined as (f + g)(n) = f (n) + g(n), likewise for the product and exponentiation. Also, we use natural numbers (and even ordinal numbers) to denote constant functions with domain ω. We denote by f + and log f the functions from ω to ω defined by: For any set A, id A denotes the identity function on A.
Given a function b with domain ω such that b(i) = ∅ for all i < ω, h ∈ ω ω and n < ω, define Depending on the context, for a set A we denote S <ω (A, h) = S <ω (b, h) where b : ω → {A}. Similarly, we use S n (A, h) and S(A, h).
Denote b := i<ω b(i) and seq <ω (b) := n<ω i<n b(i). As a topological space, we endow b with the product topology where each b(i) has the discrete topology. Note that the sets of the form [s] := [s] b := {x ∈ b : s ⊆ x} form a basis of this topology.
In particular, if each b(i) is countable then b is a Polish space, and it is perfect iff ∃ ∞ i < ω(|b(i)| ≥ 2).
2.1. Relational systems and cardinal invariants. Many of the classical cardinal invariants can be expressed by relational systems, and inequalities between these cardinals are induced by the Tukey-Galois order between the corresponding relational systems. These notions where defined by Votjas [Voj93].
Definition 2.1. A relational system is a triple A = A − , A + , < consisting of two nonempty sets A + , A − and a binary relation For x ∈ A − and y ∈ A + , x < y is often read y <-dominates x. A family X ⊆ A − is A-bounded if there is a member of A + that <-dominates every member of X, otherwise we say that the set is A-unbounded. On the other hand, Many classical cardinal invariants can be expressed through relational systems.
(3) Let b be a function with domain ω such that b(i) = ∅ for all i < ω. Define the relation = * on b as x = * y iff ∀ ∞ i < ω(x(i) = y(i)). This means that x and y are eventually different. Denote its negation by = ∞ . Put Ed(b) := b, b, = * (Ed stands for eventually different). It is clear that Functions ϕ : ω → [ω] <ω are often called slaloms.
(2) Let b be a function with domain ω such that b(i) = ∅ for all i < ω, and let h ∈ ω ω . ( , to which we often refer to as localization cardinals. (2.2) Define aLc(b, h) := S(b, h), b, * (aLc stands for anti-localization), which is a relational system. Note that , to which we refer to as anti-localization cardinals.
Goldstern and Shelah [GS93], and Kellner and Shelah [KS09,KS12] have studied cardinal coefficients of the form d Lc b,h and b aLc b,h for b ∈ ω ω (with a different notation in their work) to provide the first examples of models where continuum many cardinal characteristics of the continuum are pairwise different. On the other hand, Brendle and Mejía [BM14] investigated cardinals of the form b Lc b,h for b ∈ ω ω in relation with gap numbers for F σ -ideals. In this paper we also look at When b is the constant function ω, the localization and anti-localization cardinals provide the following well-known characterizations of classical cardinal invariants.
Hence, both anti-localization cardinals for b, h are defined iff h(i) < |b(i)| for all but finitely many i < ω, and Example 2.9. (1) Let I ⊆ J be two downwards ⊆-closed families of subsets of a nonempty set X such that ∅ ∈ I. Clearly, X, J , ∈ T X, I, ∈ . In particular, cov(J ) ≤ cov(I) and non(I) ≤ non(J ).
(2) Let b be a sequence of length ω of non-empty sets and h ∈ ω ω such that h ≥ * 1. Define (3) Let b, b be sequences of length ω of non-empty sets, and let h, h ∈ ω ω . If ∀ ∞ i < ω(|b(i)| ≤ |b (i)|) and h ≤ * h then Lc(b, h) T Lc(b , h ) and aLc(b , h ) T aLc(b, h). (4) Let b and h be as in (2). If ι : ω → ω is a one-to-one function, then ,h , likewise for the anti-localization cardinals.
Even more, if ω ranι is finite then ( The following result describes the effect of the conjunction and sequential composition on the corresponding cardinal invariants. Also define h σ ∈ ω ω such that h σ (i) = |σ(i)| for each i < ω.
Definition 2.13 (Yorioka [Yor02]). For each f ∈ ω ω define the families Any family of the form I f with f increasing is called a Yorioka ideal.
It is clear that both J g and I f contain all the finite subsets of 2 ω and that they are On the other hand, J f ⊆ N iff the series i<ω 2 −f (i) converges, and SN ⊆ J f . Hence SN ⊆ I f ⊆ N when f is increasing, so cov(N ) ≤ cov(I f ) ≤ cov(SN ) and non(SN ) ≤ non(I f ) ≤ non(N ).
In contrast, Kamo and Osuga [KO08] proved that J f is not closed under unions when f (i + 1) − f (i) ≥ 3 for all but finitely many i < ω.
Denote ω ↑ω := {d ∈ ω ω : d(0) = 0 and d is increasing}. For d ∈ ω ↑ω and an increasing Lemma 2.15 (Osuga [Osu06]). For each d ∈ ω ↑ω , g f d is increasing and g f d f for each d ∈ ω ↑ω . Conversely, for each g f there exists a d ∈ ω ↑ω such that g f d ≤ * g. In particular, if D ⊆ ω ↑ω is a dominating family, then Lemma 2.16. If f ∈ ω ω is increasing and c ∈ ω then f g iff f + c g for all g ∈ ω ω . In particular I f = I f +c .
Proof. Assume f g. Fix a natural number k ≥ 1 and choose m > c + 2 such that ∀n ≥ m(f (n k+1 ) ≤ g(n)). As f is increasing, for all n ≥ m,

2.3.
Forcing. The basics of forcing can be found in [Jec03], [Kec95] and [Kun80]. See also [BJ95] for further information about Suslin ccc forcing. Unless otherwise stated, we denote the ground model by V . When dealing with an iteration over a model V , we denote by V α the generic extension at the α-th stage.
Recall the following stronger versions of the countable chain condition of a poset.
Definition 2.17. Let P be a forcing notion and κ an infinite cardinal.
(1) For n < ω, B ⊆ P is n-linked if, for every F ⊆ B of size ≤ n, ∃q ∈ P∀p ∈ F (q ≤ p).
(2) C ⊆ P is centered if it is n-linked for every n < ω.
The following generalization of the notion of σ-linkedness is fundamental in this work.
Proof. Suppose that P = n<ω P n where each P n is centered. For each n ∈ ω, define Q n,j = P j for j < ρ(n). It is clear that Q n,j : n < ω, j < h(n) satisfies (i) and (ii) of Definition 2.18.
To fix some notation, for each set Ω let C Ω be the finite support product of Cohen forcing C := ω <ω (ordered by end-extension) indexed by Ω; D is Hechler forcing, i.e. the standard σ-centered poset that adds a dominating real; and 1 denotes the trivial poset. These three posets are Suslin ccc forcing notions.
The following poset was defined by Kamo and Osuga to increase the cardinal b aLc b,h .
Definition 2.21 (Kamo and Osuga [KO14]). Let b, h ∈ ω ω such that b ≥ 1 and assume lim i→+∞ h(i) b(i) = 0. Define the (b, h)-eventually different real forcing E h b as the poset whose conditions are of the form (s, F ) and satisfy: The following result is a generalization of [KO14, Lemma 9] about the linkedness of Let π, ρ ∈ ω ω and assume that there is a nondecreasing function f ∈ ω ω going to infinity and an m * < ω such that, for all but finitely many k < ω, Fix M > m * such that, for all k ≥ M , (i) and (ii) holds and f (k) > 0. Find a non-decreasing function g : ω → ω that goes to infinity such that, for all k ≥ M , It remains to show that Q k,s : k < ω, s ∈ S k satisfies (ii') of Definition 2.18.
, thus we can extend t to an s ∈ S k so that (s, F ) is a condition stronger than (t , F ).
. The conclusion is a consequence of Lemmas 2.20 and 2.22.
To finish this section, we review from [BM14] is σ-linked and it generically adds a slalom in S(b, h) that localizes all reals in R. In particular, LC h b generically adds a slalom in S(b, h) that localizes all the ground model reals in b.
Lemma 2.27 (Brendle and Mejía [BM14, Lemma 5.10]). Let b, h, π, ρ ∈ ω ω be nondecreasing functions with b ≥ 1 and h going to infinity. If {m k } k<ω is a non-decreasing sequence of natural numbers that goes to infinity and, for all but finitely many k < ω,

ZFC results
In this section we prove and review some inequalities between the cardinal invariants associated with Yorioka ideals, the cardinals in Cichoń's diagram and localization and antilocalization cardinals. Figure 2 at the end of this section, which is taken from [Osu08], illustrates some of these inequalities.

Localization and anti-localization cardinals. For this subsection, fix a function
. We show that the localization and anti-localization cardinals are characterized by any other known cardinal invariants when b(i) is infinite for infinitely many i < ω. We also show some results for these cardinals when b ∈ ω ω , mostly when taking limit values like sup{d Lc b,h : b ∈ ω ω }. When h does not go to infinity, the localization cardinals have simple characterizations.
On the other hand, [GS93, Lemma 1.11] states that d Lc b,h = c whenever h is constant and b ∈ ω ω . So, by Example 2.9(3) and (4), d Lc b,h ≥ c holds in our case. In addition, if so the "moreover" part follows by Example 2.9(3).
A similar result to the above can be proved for the anti-localization cardinals when the On the other hand, by Example 2.9(2) and Theorem 3.
Proof. We first show (a). Wlog, we may assume that b(i) is an infinite cardinal for all i < ω. Find a non-decreasing function b 0 from ω into the infinite cardinals such that b 0 (i) ≤ b(i) for all i < ω, and sup i<ω {b 0 (i)} = κ. By Example 2.9(3) and Theorem 2.5, To see that min{add has size less than this minimum and show that there is some ,1 (see Theorem 3.1 and Corollary 3.6(b)) and, although b Lc b,h depends on h, it is a finite number already calculated in Theorem 3.1.
When b is above the continuum, b aLc b,h and d Lc b,h are simpler to calculate.
It is a direct consequence of Theorem 3.4 and the fact that cf( The cases that are not characterized in Theorem 3.4 are when b(i) is finite for all (but finitely many) i < ω for the localization cardinals, and when b(i) is finite for infinitely many i < ω for the anti-localization cardinals. Even more, by the following result (for the case κ = ω), the latter case is reduced to the case when b(i) is finite for all i < ω.
Lemma 3.7. Let κ be an infinite cardinal. If the set F := {i < ω : |b(i)| < κ} is infinite by Example 2.9(3) and Theorem 3.4. In a similar way, Hence, the result follows by Theorem 2.11. Now, we look at limits of localization and anti-localization cardinals.
Definition 3.8. Define the following cardinal characteristics.
To see (a) it is enough to show the following.
On the other hand, minLc h ≤ minLc h and supLc h ≤ supLc h by Example 2.9(3), so equality holds. Therefore, (a) follows by using an h above both id ω and h.
Concerning item (b), we have Proof. Let I n : n < ω be the interval partition of ω such that |I n | = h(n) for all n < ω.
On the other hand, define ϕ + : b → b by ϕ + (y) := y I n : n < ω . It is clear that (ϕ − , ϕ + ) is the Tukey connection we want.
As in the proof of (a), the claim above can be used to prove (b).
Other cardinals like sup{b Lc b,h : b ∈ ω ω } are in principle not that interesting, for instance, this supremum above would be b Lc 1,h , which is undefined, or at least if b is restricted to be above h, then it would be b Lc h+1,h . Also, when h does not go to infinity, minLc h and supLc h are easily characterized by Theorem 3.1.
The following is another characterization of add(N ) and cof(N ) in terms of Localization cardinals for b ∈ ω ω . Lemma 3.12. add(N ) = min{b, minLc} and cof(N ) = max{d, supLc}.
Proof. In this proof we use the characterization of add(N ) and cof(N ) given in Theorem 2.4 Assume that F ⊆ ω ω and |F | < min{b, minLc}. Therefore, there is some d ∈ ω ω such that, for every x ∈ F , ∀ ∞ i < ω(x(i) < d(i)). On the other hand, as |F | < b Lc d,idω , we can find an slalom in S(d, id ω ) ⊆ S(ω, id ω ) that localizes all the reals in F (just use a family F ⊆ d of the same size as F such that each member of F is a finite modification of some member of F and viceversa). Therefore, add(N ) ≥ min{b, minLc}). Now we prove cof(N ) ≤ max{d, supLc}. Choose a dominating family D ⊆ ω ω of size d and, for each d ∈ D, choose a family of slaloms S d ⊆ S(d, id ω ) that witnesses d Lc d,idω . Note that S := d∈D S d ⊆ S(ω, id ω ) has size ≤ max{d, supLc} and that every real in ω ω is localized by some slalom in S, so cof(N ) ≤ |S|.
It is easy to see that D T Lc(ω, id ω ), so add(N ) ≤ b and d ≤ cof(N ). On the other hand, as add(N ) ≤ b Lc b,idω and d Lc b,idω ≤ cof(N ) for any b ∈ ω ω (see Example 2.9(3)), the converse inequalities follow.
The version of this lemma for the anti-localization cardinals is Miller's [Mil81] known result add(M) = min{b, minaLc} and its dual cof(M) = max{d, supaLc}, which is proved in Theorem 3.24. Miller also proved that non(SN ) = minaLc. At the end of Section 5 we explain why no inequality between each pair of these cardinals can be proved in ZFC .
3.2. Additivity and cofinality of Yorioka ideals. Though Corollary 3.14 and Theorem 3.16 are stated in [Osu08], their proofs seem not to appear in any existing reference. We offer original proofs of both results, even more, we provide the following general result that gives us Corollary 3.14 as a direct consequence. This is also used in Section 5 to prove that, consistently, add(N ) < add(I f ) < cof(I f ) < cof(N ).
Proof. Apply Theorem 3.13 with b = ω and h = id ω .
In the previous result we did not use the fact that I f is closed under unions, so it implies that I f is a σ-ideal (see Theorem 2.14).
where m is the minimum natural number such that g f e * (n) ≤ g f e (m). Clearly, m n, so |ρ Y,e (n)| ≤ g f e (n).
5 Intuitively, ρ Y,e (n) is the sequence that results by cutting from τ Y,e (n) the blocks that are indexed by [g f e (k − 1) + ∆g f e (k), g f e (k)) (for k < ω). 6 Intuitively, x results by inserting (infinitely many) blocks of 0's inside x like in the definition of τ X,e .

Lemma 3.19 (Kamo and Osuga
Theorem 3.21. supcov = supaLc and minnon = minaLc. Proof. We first prove that, for all b ∈ ω ω , b aLc b,1 supcov and minnon ≤ d aLc b,1 . Wlog b ≥ 2 (By Example 2.9(3)). Put f = (log b) + . As J f ⊇ I f , cov(J f ) cov(I f ) and non(I f ) non(J f ). By Lemma 3.19, b aLc b,1 ≤ cov(J f ) and non(J f ) ≤ d aLc b,1 . To prove the converse inequalities, assume that f ∈ ω ω is increasing. Choose some g f monotone increasing, so J g ⊆ I f . Hence cov(I f ) cov(J g ) and non(J g ) ≤ non(I f ). Put b = 2 g . By Lemma 3.20, cov(J g ) ≤ b aLc b,1 and d aLc b,1 ≤ non(J g ), so cov(I f ) ≤ b aLc b,1 ≤ supaLc and minaLc ≤ d aLc b,1 ≤ non(I f ). On the other hand, by Example 2.9(3) and the previous result, we have We know that add(M) = min{b, cov(M)} and cof(M) = max{d, non(M)}, but these equalities can be refined as in the following two results. These yield a version of Lemma 3.12 for the anti-localization cardinals.  Proof. The inequality ≥ follows from Corollary 3.22.

Preservation properties
The preservation properties discussed in this section were developed for FS iterations of ccc posets by Judah-Shelah [JS90] and Brendle [Bre91], later generalized and summarized in [BJ95], [Mej13] and [FFMM18]. We generalize this theory so that preservation properties as in [KO14] (see Example 4.20) become particular cases. Afterwards, we show how to adapt this theory to preserve unbounded reals along FS iterations, which is useful in the context of matrix iterations. 4.1. The presevation theory. Our notation is closer to [FFMM18]. The classical preservation theory of Judah-Shelah and Brendle corresponds to the case |Ω| = 1 of the definition below. Though the proofs of the facts in this section follow the same ideas as the classical results, the arguments are presented for completeness.
Definition 4.1. Say that R = X, Y, < is a generalized Polish relational system (gPrs) if (I) X is a Perfect Polish space, (II) Y = e∈Ω Y e where Ω is a non-empty set and, for some Polish space Z, Y e is non-empty and analytic in Z for all e ∈ Ω, and (III) <= n<ω < n where < n : n < ω is some increasing sequence of closed subsets of X × Z such that, for any n < ω and for any y ∈ Y , (< n ) y = {x ∈ X : x < n y} is closed nowhere dense. If |Ω| = 1, we just say that R is a Polish relational system (Prs). For a set A and x ∈ X say that x is R-unbounded over A if ∀y ∈ A ∩ Y (x < y).
For θ ≥ 2, any θ-R-unbounded family is R-unbounded and, for θ regular, if F is a strongly θ-R-unbounded family then it is |F |-R-unbounded. In consequence, The following are useful properties to preserve (strongly) θ-R-unbounded families in forcing generic extensions. In this context, X, Z and < are interpreted in transitive models of ZFC as Polish spaces, while Y is interpreted as Y M = e∈Ω Y M e for such a model M containing the information to define Y . As in the case of Polish spaces, we also omit the upper indices M on the interpretation of Y .
Definition 4.5. Let P be a forcing notion and θ a cardinal.
(1) P is θ-R-good if, for any P-nameḣ for a member of Y , there exists a non-empty H ⊆ Y (in the ground model) of size < θ such that, for any x ∈ X, if x is R-unbounded over H then x <ḣ.
(2) P is θ-R-nice if, for all e ∈ Ω and for any P-nameḣ for a member of Y e , there exists a non-empty H ⊆ Y of size < θ such that, for any x ∈ X, if x is R-unbounded over H then x <ḣ. Say that P is R-good (R-nice) if it is ℵ 1 -R-good (ℵ 1 -R-nice).
Note that θ < θ implies that any θ-R-good poset is θ -R-good. Also, if P Q and Q is θ-R-good, then P is θ-R-good. Similar facts hold for niceness. It is clear that any θ-R-good forcing notion is θ-R-nice. The converse holds in some cases as below.
Lemma 4.6. Let θ be a regular cardinal. If either P is θ-cc or |Ω| < θ, then P is θ-R-nice iff it is θ-R-good.
Proof. Assume that either P is θ-cc or |Ω| < θ. Letḣ an P-name for a member of Y . Choose a maximal antichain A in P and {e p : p ∈ A} ⊆ Ω such that p ḣ ∈ Y ep for all p ∈ A. Put Γ := {e p : p ∈ A}. By hypothesis, Γ has size < θ. For each β ∈ Γ define A β = {p ∈ A : e p = β}. As p ḣ ∈ Y β for any p ∈ A β , we can find a P-nameẏ β of a member of Y β such that p ḣ =ẏ β for any p ∈ A β . As P is θ-R-nice, for each β ∈ Γ there exists a non-empty H β ⊆ Y of size < θ that witnesses niceness forẏ β . Put H = β∈Γ H β which has size < θ because θ is regular. Assume that x ∈ X is R-unbounded over H. Given p ∈ A, there is a β ∈ Γ such that p ∈ A β . As x ∈ X is R-unbounded over H β , p x <ẏ β and, on the other hand, p ḣ =ẏ β , so p x <ḣ. As A is a maximal antichain, x <ḣ.
Lemma 4.7. Let θ be a regular cardinal, λ ≥ θ a cardinal and let P be a θ-R-good poset. (a) If F ⊆ X is λ-R-unbounded, then P forces that it isλ -R-unbounded where, in the P-extension,λ is the smallest cardinal ≥ λ. (b) If cf(λ) ≥ θ and F ⊆ X is strongly λ-R-unbounded then "if λ is a cardinal then F is strongly λ-R-unbounded". (c) If d(R) ≥ λ then P forces that d(R) ≥λ .
Proof. (a) It is enough to consider sets of P-names for members of Y of the form {ġ α } α<η for some η < λ. For α < η, let H α ⊆ Y of size < θ that witnesses the goodness of P forġ α . Put H = α<η H α . As |H| < λ, there is some x ∈ F R-unbounded over H. Thus x <ġ α for any α < η. (b) Repeat the argument above with η = 1 and find H. Hence {x ∈ F : x <ġ 0 } ⊆ h∈H {x ∈ F : x < h}. That union has size < λ in the ground model. On the other hand, F is forced to have size ≥ |λ|. (c) Consequence of (a) because V |="X is λ-R-unbounded".
We now aim to prove that θ-R-goodness is respected in FS iterations of θ-cc posets.
Definition 4.8. Let P be a forcing notion and letż be a P-name for a real in ω ω . A pair ( p n n<ω , g) is called an interpretation ofż in P if g ∈ ω ω and, for all n < ω, (i) p n ∈ P, p n+1 ≤ p n , and (ii) p n ż n = g n. Say that this interpretation is below p ∈ P if, additionally, p 0 ≤ p. Lemma 4.9. Assume that P is a poset , e ∈ Ω, f : ω ω → Y e is a continuous function, z is a P-name for a real in ω ω and ( p n n<ω , g) is an interpretation ofż in P. If x ∈ X, n < ω and x < n f (g), then there is a k < ω such that p k x < n f (ż).
Proof. As {y ∈ Y e : x < n y} is closed in Y e (see Definition 4.1) and f : Lemma 4.10. If θ is a cardinal then any poset of size < θ is θ-R-nice. Moreover, if θ is regular then any such poset is θ-R-good. In particular, C is R-good.
The 'moreover' part follows by Lemma 4.6.
Lemma 4.11. Let θ be a regular cardinal, P a poset andQ a P-name for a poset. If P is θ-cc, θ-R-good and it forces thatQ is θ-R-good, then P * Q is θ-R-good Proof. Letḣ be a P * Q-name for a member of Y . Wlog P forces thatḣ is aQ-name for a member for Y . As P forces thatQ is θ-R-good, P forces that there is a nonemptyḢ ⊆ Y of size < θ such that, for any x ∈ X, if x R-unbounded overḢ, then Q x <ḣ. As P is θ-cc, we can find ν < θ in the ground model so thatḢ = {ẏ α : α < ν}. For each α < ν let B α be a witness of goodness of P forẏ α . Put B := α<ν B α , which has size < θ. It is easy to see that, if x ∈ X is R-unbounded over B, then P * Q x <ḣ.
We show that goodness is preserved along direct limits in quite a general way so that the theory of this section can be applied to template iterations as in [Mej15,Sect. 5]. Say that a partial order I is directed if, for any i, j ∈ I, there is a k ∈ I such that i, j ≤ k. A system P i i∈I of posets indexed by a directed partial order I is called a directed system of posets if P i is a complete subposet of P j for all i ≤ j in I. For such a directed system, define its direct limit by limdir i∈I P i := i∈I P i .
Theorem 4.12. Let θ be a regular cardinal, P i i∈I a directed system of posets and P := limdir i∈I P i . If |I| < θ and P i is θ-R-nice for all i ∈ I, then P is θ-R-nice.
Proof. Let e ∈ Ω and letḣ be a P-name for member in Y e . Choose a continuous and surjective function f : ω ω → Y e and a P-name for a realż in ω ω such that P forces that f (ż) =ḣ. For each i ∈ I, find a P i -name for a realż i in ω ω and a sequence ṗ i,k k<ω of P i -names such that P i forces that ( ṗ i,k k<ω ,ż i ) is an interpretation ofż in P/P i . Choose H i ⊆ Y of size < θ such that it witnesses goodness of P i for f (ż i ). Put H = i∈I H i , which has size < θ since |I| < θ and θ is regular.
We prove that, if x ∈ X is R-unbounded over H, then P x <ḣ. Assume towards a contradiction that there are p ∈ P and n < ω such that p P x < nḣ . Choose i ∈ I such that p ∈ P i . Let G be a P i -generic over the ground model V with p ∈ G. By the choice of ). By Lemma 4.9, there is a k < ω such thaṫ p i,k [G] P/P i x < n f (ż) =ḣ. On the other hand, by hypothesis, p P i " P/P i x < nḣ ", a contradiction.
Remark 4.13. We can replace the hypothesis |I| < θ by cf(I) < θ in the previous result. This is because limdir i∈I P i = limdir i∈C P i for any cofinal C ⊆ I.
Proof. We prove that P α is θ-R-good by induction on α ≤ δ. The step α = 0 follows by Lemma 4.10 and the successor step follows by Lemma 4.11. Assume that α is a limit ordinal. If cf(α) < θ then P α is θ-R-good by Theorem 4.12 and Lemma 4.6.
Assume that cf(α) ≥ θ. Letḣ be a P α -name for a member of Y . By θ-cc-ness, there exists a ξ < α such thatḣ is a P ξ -name. As P ξ is θ-R-good, there is a non-empty H ⊆ Y of size < θ that witnesses goodness of P ξ forḣ. It is clear that H also witnesses goodness of P α forḣ.
Recall that c ∈ X is a Cohen real over a model M if c does not belong to any Borel meager set coded in M . It is clear that Cohen forcing adds such a real over the ground model. Indeed, given a metric d on ω such that X, as a complete metric space, is a completion of ω, d , consider C d := {t ∈ ω <ω : ∀i < |t| − 1(d(t i , t i+1 ) < 2 −(i+2) )} ordered by end-extension. This is a countable atomless poset (because ω, d is perfect), so it is forcing equivalent to C. It is not hard to see that C d adds a Cauchy-sequence that converges to a Cohen real in X over the ground model. By Definition 4.1(III), any Cohen real in X over the ground model is R-unbounded over the ground model. Hence, it is possible to add (strongly) R-unbounded families with Cohen reals through FS iterations.
Lemma 4.15. If ν is a cardinal with uncountable cofinality and P ν = P α ,Q α α<ν is a FS iteration of non-trivial cf(ν)-cc posets, then P ν adds a strongly ν-R-unbounded family of size ν.
Proof. The Cohen reals (in X) added at the limit steps of the iteration form a strongly ν-R-unbounded family of size ν.
Proof. (a) is a direct consequence of Lemmas 4.7, 4.15 and the fact that P δ /P ν , the remaining part of the iteration from stage ν, is θ-R-good (by Corollary 4.14). On the other hand, by Lemma 4.4, b(R) ≤ θ follows from (a) for ν = θ and d(R) ≥ |δ| follows from (b) for λ = |δ|.
It remains to prove (b) for the case when λ is singular (for λ regular it just follows from (a)). Work in V λ = V P λ . Let {δ ξ : ξ < λ} be the increasing enumeration of 0 and the limit ordinals below λ and, for each ξ < λ, denote by c ξ the Cohen real in X over V δ ξ added by P δ ξ+1 . As explained in the proof of Lemma 4.15, for each ν ∈ [θ, λ) regular, {c ξ : ξ < ν} ∈ V ν is (strongly) ν-R-unbounded in V ν , and also in V λ by (a). Thus {c ξ : ξ < λ} is λ-R-unbounded. Indeed, if A ⊆ Y has size < λ then it has size < ν for some regular ν ∈ [θ, λ), so there is some ξ < ν such that c ξ is R-unbounded over A.
When h ≥ * 1 and b > * h e for any e ∈ E, it is clear that Lc  *  (b, h) is a gPrs.
By the previous result, we can use the preservation property of aLc * (b, h) to decide the values of b aLc b,h idω and d aLc b,h idω by forcing, likewise for Lc * (b, h) and the localization cardinals. Lemma 4.23. Let n < ω and B ⊆ P be n-linked. If F ∈ V has size ≤ n andȧ is a P-name for a member of F , then there exists a c ∈ F such that no p ∈ B forcesȧ = c.
Proof. This was proved for the gPrs aLc * (b, h) in [KO14, Lemma 10]. The case of Lc * (b, h) follows from the same proof, which we include for completeness. Let P be a poset and let Q n,j : n < ω, j < h(n) be a sequence that witnesses that P is (h, b h idω )-linked. Assume thatφ is a P-name of a member ofŜ(b, h). As P is ccc (by Lemma 2.20), we can find an e ∈ E such that P forces that ϕ ∈ S(b, h e ). Furthermore, choose e ∈ E and N < ω such that h(n) · e(n) > 0 and e(n) + 1 ≤ e (n) for all n ≥ N , so we can find a P-nameφ of a member of S(b, h e ) such that P forcesφ(n) ⊆φ (n) = ∅ for all n ≥ N .
For each n ≥ N and j < h(n), as Q n,j is b(n) h(n) n -linked and [b(n)] ≤h(n) e(n) {∅} has size ≤ b(n) h(n) n , by Lemma 4.23 there is an a n,j ⊆ b(n) of size ≤ h(n) e(n) such that p a n,j = ϕ (n) for all p ∈ Q n,j . Note that ψ(n) := j<h(n) a n,j has size ≤ h(n) e(n)+1 ≤ h(n) e (n) , which gives us a ψ ∈ S(b, h e ). It is clear that (i) if ϑ ∈ S(b, h idω ) and ϑ ψ then ϑ φ (which implies ϑ φ), and (ii) if x ∈ b and ¬(x ∈ * ψ) then ¬(x ∈ * φ ) (so ¬(x ∈ * φ )). This concludes the proof.
Proof. The latter part is a consequence of Lemma 4.6.
The following example abbreviates many facts about the main preservation result in [BM14]. This will not be used in any other part of this text.
For each k < ω define the relation  For the following results, fix a gPrs R = X, Y, < coded in M (in the sense that all its components are coded in M ). We are interested in preserving R-unbounded reals between forcing extensions of M and N .
Proof. Let Z be the Polish space where S is defined, and recall the Polish space Z that contains Y (see Definition 4.1). Choose a metric space η, d with η ≤ ω such that Z, as a complete metric space with metric d * , is a completion of η, d . Note that any (good) S-name of a member of Z can be seen as a name of a Cauchy sequence k m : m < ω in η, d such that d(k m ,k m+1 ) < 2 −(m+2) for all m < ω. This can be coded by a member of (Z × η) ω×ω . Therefore, "ḣ is a S-name of a member of Z" is a conjunction between a Σ 1 1 -statement and a Π 1 1 -statement in (Z × η) ω×ω . Indeed, a S-nameḣ of a member of Z is a functionḣ = (p˙h, k˙h) : ω × ω → Z × η such that, for each m < ω, (i) {p˙h(m, n) : n < ω} is a maximal antichain in S (the point is that p˙h(m, n) decides that the m-th term of the Cauchy-sequence in η, d converging toḣ is k˙h(m, n)), and (ii) for each n, n < ω, if p˙h(m, n) and p˙h(m + 1, n ) are compatible in S then d(k˙h(m, n), k˙h(m + 1, n )) < 2 −(m+2) (which means thatḣ is a name of a Cauchy-sequence as described before). The statement in (i) can be expressed as a conjunction between a Σ 1 1 -statement and a Π 1 1 -statement, while (ii) is a Π 1 1 -statement. Even more, if S is a Borel subset of Z then (i) is a Π 1 1 -statement. Claim 4.29 ([Mej13, Claim 1]). The statement Ψ(ḣ,ȳ) that says "ḣ is a S-name for a member of Z and, for all x ∈ X, if x < y n for each n < ω then S x <ḣ" is a conjunction of a Σ 1 1 -statement with a Π 1 1 -statement in (Z × η) ω×ω × Z ω . Even more, if S is Borel in Z , then the statement is Π 1 1 . Proof. It is just enough to look at the complexity of " S x <ḣ". This is equivalent to say that "for every p ∈ S and l < ω there are positive rational numbers r, ε and k, m, n < ω such that p is compatible with p˙h(m, n), B(k, r) ∩ {z ∈ Z : x < l z} = ∅ and d(k, k˙h(m, n)) < r − 2 −(m+1) − ε" where B(k, r) := {z ∈ Z : d * (k, z) < r}. This statement can be written in the form ∀p ∈ Z(p / ∈ S or Θ(p, x,ḣ)) where Θ(p, x,ḣ) is Π 1 1 (recall that compatibility in S is a Π 1 1 -relation in Z and that "B(k, r) ∩ {z ∈ Z : x < l z} = ∅" is also Π 1 1 ). Hence, as "p ∈ S" is Σ 1 1 , the whole statement is Π 1 1 . On the other hand, as discussed before the claim, "ḣ is a S-name for a member of Z" is a conjunction of a Σ 1 1 -statement with a Π 1 1 -statement. In M , fix a S-nameḣ for a real in Y and a countable H ⊆ Y that witnesses the goodness of S forḣ. Enumerate H = {y n : n < ω}. Now, as Ψ(ḣ, y n : n < ω ) is true in M , it is also true in N by Claim 4.29 and Π 1 1 -absoluteness. In N , as c is R-unbounded over M , ∀n < ω(c < y n ), so N S c <ḣ. Lemma 4.30 ([BF11, Lemma 11]). Assume that P ∈ M is a poset. Then, in N , P forces that every c ∈ X N that is R-unbounded over M is R-unbounded over M P .
Proof. Work within M . Let e ∈ Ω andḣ be a P-name for a member of Y e . Fix p ∈ P and n < ω. Choose a continuous and surjective function f : ω ω → Y e and a P-nameż for a real in ω ω such that P forces that f (ż) =ḣ. Choose an interpretation ( p k k<ω , g) ofż below p. In N , as c is R-unbounded over M , then c < f (g), so c < n f (g). By Lemma 4.9, there is a k < ω such that p k N P c < n f (ż) =ḣ. Let P 0 , P 1 , Q 0 , Q 1 be partial orders with P 0 , P 1 ∈ M . Recall that P 0 , P 1 , Q 0 , Q 1 is correct with respect to M if P 0 is a complete subposet of P 1 , Q 0 is a complete subposet of Q 1 , P i M Q i for each i < 2 and, whenever p 0 ∈ P 0 is a reduction of p 1 ∈ P 1 , then p 0 is a reduction of p 1 with respect to Q 0 , Q 1 (see [Mej15,Def. 2.8]). that f (ż) =ḣ. Work in N . Assume, towards a contradiction, that there are q ∈ Q and n < ω such that q N Q c < nḣ . Choose i ∈ I such that q ∈ Q i . Let G be Q i -generic over N such that q ∈ G. By assumption, Q/Q i c < nḣ . In M [G ∩ P i ], find and interpretation ( p k k<ω , g) ofż in P/P i . In N [G], as c < f (g), by Lemma 4.9 there is a k < ω such that p k Corollary 4.32. Let δ be an ordinal, c ∈ X N , P 0 α ,Q 0 α α<δ ∈ M and P 1 α ,Q 1 α α<δ ∈ N both FS iterations such that, for any α < δ, if P 0 α M P 1 α then, in N , P 1 α forces thaṫ Q 0 α M P 0 αQ 1 α and thatQ 1 α forces that c is R-unbounded over M P 0 α+1 . Then, P 0 α M P 1 α for all α ≤ δ and P 1 δ forces, in N , that c is R-unbounded over M P 0 δ . Proof. Fix α ≤ δ. When P 0 α M P 1 α it is not hard to see by induction on β ∈ [α, δ] that P 0 α , P 0 β , P 1 α , P 1 β is correct with respect to M (the limit step follows by Lemma 4.31, the successor step by [BF11,Lemma 13]). The result follows by the case α = 0 and by Lemma 4.31.

Consistency results
Before we prove the consistency results of this section, we review some facts about matrix iterations, including one result about preservation in the context of general Polish relational systems.
The construction of the matrix iterations in our consistency results corresponds to the following particular case.
it consists, additionally, of (i) a partition S m , T m of [1, δ m ), (ii) a function ∆ m : T m → {α ≤ γ m : α is not limit}, (iii) a sequence S m ξ : ξ ∈ S m of Suslin ccc posets coded in the ground model V , (iv) a sequence Ṫ m ξ : ξ ∈ T m such that eachṪ m ξ is a P m ∆ m (ξ),ξ -name of a poset such that it is forced by P m γ,ξ to be ccc, and (III) for each α ≤ γ and 1 ≤ ξ < δ, otherwise.
In practice, P m α,ξ : α ≤ γ is constructed by induction on ξ ≤ δ (before the ξ + 1-th step for ξ ∈ T m ,Ṫ m ξ is defined). So it is clear that the constructed system m is a matrix iteration. Moreover, each P m α,ξ is ccc. Again, when there is no place for confusion, we omit the upper index m.
To finish the proof it remains to show that κ ≤ minnon. For any b ∈ ω ω , as D is σ-centered and thus aLc * (b , 1)-good by (P4), κ ≤ d(aLc * (b , 1)) ≤ d aLc b ,1 by Theorem 5.4 and (P6). Therefore, by Theorem 3.21, κ ≤ minnon = min{d aLc b ,1 : b ∈ ω ω }. Remark 5.9. If the matrix iteration construction of Theorem 5.7 is modified so that S m = ∅ (i.e. no use of Hechler forcing) then the cardinal invariants associated with many Yorioka ideals can still be separated. Concretely, the final model still satisfies (A), (B1) and (B2), and also satisfies cov(I f ) = non(M) = ν and cov(M) = non(I f ) = κ for all increasing f ≥ * f π . However, the values of b and d are unknown because it is unclear whether the restricted versions of LC c ζ ,h ζ and E c ζ (ζ ≤ π) used in the construction add dominating reals.
Remark 5.10. In the hypothesis of Theorem 5.7 assume, additionally, that µ is a regular cardinal and µ ≤ µ ≤ ν and, instead of κ being regular, just assume that κ <µ = κ. As in [Mej], the forcing construction in Theorem 5.7 can be modified so that the matrix iteration allows vertical support restrictions, that is, P A,ξ can be defined for all A ⊆ κ and ξ ≤ π. The final model of this construction still satisfies (A), (B1) and (B2), and also satisfies cov(I f ) = supcov = µ , add(M) = cof(M) = ν and minnon = non(I f ) = κ (the latter not necessarily regular) for all increasing f ≥ * f π . Though this is stronger than Theorem 5.7, we do not get to separate more of the cardinal invariants associated with Yorioka ideals.
Remark 5.11. In the context of [BM14], Theorem 5.7 could be modified so that, in (B1), it can be forced that b Lc c ζ ,h ζ = b(I ζ ) = θ ζ for ζ < ζ * where I ζ : ζ < ζ * is some sequence of gradually fragmented ideals on ω and b(I ζ ) denotes the Rothberger number of I ζ .
The next result guarantees the consistency of add(N ) < add(I f ) < cof(I f ) < cof(N ) for any fixed f .
Claim 5.13. In V κ,λκν , each family of reals of size < ν is dominated by some d ρ .
As a direct consequence, ν ≤ b. On the other hand, {d ρ : ρ < κν} is a family of reals of size ≤ κ and, by Claim 5.13, any member of V κ,λκν ∩ ω ω is dominated by some d ρ . Hence d ≤ κ. By a similar argument, we can prove: Claim 5.14. Each family of reals in V κ,λκν of size < ν is localized by some slalom ψ ρ .
In the previous model, it is clear that minLc ≤ b and d ≤ supLc, so add(N ) = minLc and cof(N ) = supLc by Theorem 3.12.
To finish this section, we show the consistency of b < minLc and d < supLc. As the converse strict inequalities hold in the model of Theorem 5.7, each pair of cardinals are independent. In particular, the characterizations add(N ) = min{b, minLc} and cof(N ) = max{d, supLc} (see Theorem 3.12) are optimal. Note that b ≤ minLc implies add(N ) = minadd = add(I idω ) = b, and supLc ≤ d implies cof(N ) = supcof = cof(I idω ) = d.
Proof. Construct P by a FS iteration of length λν of LC idω b for every b ∈ ω ω (including those that appear in intermediate extensions), and of all the ccc posets of size < µ with underlying set an ordinal < µ (like in the previous results). We also demand that LC idω b for each b ∈ ω ω is used cofinally often. By this construction, P is µ-D-good (by Lemma 4.10 and Example 4.17(3)) and it forces MA <µ , so it forces add(N ) = b = µ and d = c = λ by Theorem 4.16. On the other hand, it is clear that P forces b Lc b,idω = non(M) = cov(M) = d Lc b,idω = ν for any b ∈ ω ω (also thanks to the Cohen reals added at limit stages), so minLc = supLc = ν.
With respect to the pairs b, minaLc and d, supaLc, it is clear that a FS iteration of big size but short cofinality of Hechler posets forces supaLc < b and d < minaLc (b < d can be additionally forced with a matrix iteration as in Theorem 5.12 but using only (i)-(iii) for the construction). If after this iteration we force with a large random algebra, then minaLc = non(N ) = ℵ 1 < b and d < cof(N ) = supaLc is satisfied in the final extension.

Discussions and Open Questions
Though we constructed a model of ZFC where the four cardinal invariants associated with many Yorioka ideals are pairwise different, we still do not know how to construct a model where we can separate those cardinals for all Yorioka ideals.
Question 6.1. Is there a model of ZFC satisfying add(I f ) < cov(I f ) < non(I f ) < cof(I f ) for all increasing f : ω → ω?
It is not even known whether there is such a model for just f = id ω . There are many ways to attack this problem. One would be to find a gPrs R that satisfies add(I idω ) ≤ b(R) and d(R) ≤ cof(I idω ) and such that any poset of the form E b is R-good. If such a gPrs can be found, then a construction as in Theorem 5.7 works. Other way is to adapt the techniques for not adding dominating reals from [GMS16] in matrix iterations so that a construction as in Remark 5.9 could force b ≤ µ and λ ≤ d. This is quite possible because E b is D-good (see Example 4.17(3)), though its restriction to some internal model may add dominating reals (see [Paw92]). Success with this method would actually solve the following problem. Such a model exists assuming strongly compact cardinals (see [GKS]), but it is still open modulo ZFC alone.
Other way to attack Question 6.1 is by large products of creature forcing (see [KS09,KS12,FGKS17]). Quite recently, A. Fischer, Goldstern, Kellner and Shelah [FGKS17] used this technique to prove that 5 cardinal invariants on the right of Cichoń's diagram are pairwise different. Such method would also work to solve the dual of Question 6.1, that is, Question 6.3. Is there a model of add(I f ) < non(I f ) < cov(I f ) < cof(I f ) for all increasing f : ω → ω?
There are still many open questions in ZFC about the relations between the cardinal invariants associated with Yorioka ideals and the cardinals in Cichoń's diagram. For instance, it is unknown whether minadd ≤ cov(N ) (or non(N ) ≤ supcof) is provable in ZFC. In general, very little is known about the additivity and the cofinality of Yorioka ideals.
Question 6.4. Is it consistent with ZFC that infinitely many cardinal invariants of the form add(I f ) are pairwise different? Question 6.5. Is it true in ZFC that add(I f ) = add(I idω ) (and cof(I f ) = cof(I idω )) for all (or some) increasing f ? Question 6.4 is related with finding a gPrs associated to add(I f ) and cof(I f ).
Theorem 3.12 indicates that the method of Theorem 5.12 cannot be used to increase every cardinal of the form add(I f ) while preserving add(N ) small.
Concerning the problem of separating the four cardinal invariants associated with an ideal, the consistency of add(N ) < cov(N ) < non(N ) < cof(N ) is known from [Mej13].
The same question for M is also open. Though in Theorem 5.7 we were able to separate (infinitely) many localization and antilocalization cardinals, the b-localization cardinals appear below the b-anti-localization ones. The reason of this is that the preservation methods we use for the localization cardinals relies on the structure R ρ * , so after including one b-anti-localization cardinal with functions quite above ρ * (to preserve the previous localization cardinals), a new forcing to increase some other localization cardinal may not preserve the previous antilocalization ones. In view of this, it would be interesting to improve our preservation methods to allow both type of cardinals appear alternatively.