{'fraction_non_alphanumeric': 0.09066907077081589, 'fraction_numerical': 0.011805868589949985, 'mean_word_length': 3.2275862068965515, 'pattern_counts': {'":': 0, '<': 126, '<?xml version=': 0, '>': 11, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 27, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Many Different Uniformity Numbers of Yorioka Ideals

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals (localisation and anti-localisation cardinals), for uncountably many parameters the corresponding cardinals are pairwise different.


INTRODUCTION
This research forms part of the study of cardinal characteristics of the continuum which are parametrised by reals and of the forcing techniques required to separate many of them. The main motivation of this paper is to produce a forcing model where several (even uncountably many) uniformity numbers of Yorioka ideals are pairwise different; in doing so, we included additional types of parametrised cardinal characteristics which we refer to as localisation and anti-localisation cardinals. We first review the definition of a Yorioka ideal: Notation 1.1. We fix the following terminology.
(2) For f ∈ ω ω increasing, define I f := g f J g . Any family of this form is called a Yorioka ideal.
Yorioka ideals are partial approximations of the σ-ideal SN of strong measure zero subsets of 2 ω . They were introduced by Yorioka [Yor02] to show that no inequality between cof(SN ) and c := 2 ℵ 0 can be proved in ZFC. Though it is very easy to show that SN = g∈ω ω J g , the families of the form J g are not ideals in general (Kamo and Osuga [KO08]); however, I f is a σ-ideal whenever f is increasing, 1 and SN = {I f | f ∈ ω ω increasing} characterises SN , as well. Also note that I f ⊆ N , where N denotes the ideal of Lebesgue measure zero subsets of 2 ω . The cardinal invariants associated with Yorioka ideals (that is, add(I f ), cov(I f ), non(I f ) and cof(I f )) have been studied for quite some time. Kamo and Osuga [KO08] showed that for any fixed increasing f , no other inequality consistent with the standard known inequalities (see Figure 1) can be proved in ZFC. This was improved by Cardona and the second author [CM19] by constructing a ccc poset forcing that, for some f 0 , the four cardinals associated with I f are pairwise different for all f ≥ * f 0 . Very recently, the dependence on some f 0 was eliminated (see [BCM21]). On the other hand, Kamo and Osuga also showed that add(I f ) ≤ b and d ≤ cof(I f ). It is also known that add(N ) ≤ add(I f ) and cof(I f ) ≤ cof(N ) in ZFC (attributed to Kamo; see also [CM19, section 3] for a proof), but equality cannot be proved (Cardona and the second author [CM19]).
Later, Kamo and Osuga [KO14] forced, using a ccc poset, that infinitely (even uncountably) many covering numbers of Yorioka ideals are pairwise different; moreover, continuum many pairwise different covering numbers can be forced under the assumption that a weakly inaccessible cardinal exists. The dual of this result is the main question of this paper.
Question A. Is it consistent with ZFC that infinitely many cardinals of the form non(I f ) are pairwise different?
A key feature in Kamo and Osuga's model for infinitely many covering numbers is a relation they discovered between the covering numbers of Yorioka ideals and anti-localisation cardinals of the form c ∃ c,h . The duals v ∃ c,h of these cardinals, as well as the localisation cardinals themselves, will also play an important role in solving the question above. They are defined as follows: Notation 1.3. We fix some basic notation and terminology.
(5) When, in addition, x and y go into the ordinal numbers, we write x ≤ y for ∀ i < ω : x(i) ≤ y(i) and x ≤ * y for ∀ ∞ i < ω : x(i) ≤ y(i). We likewise use the notation x < y and x < * y.
Definition 1.4. Let c = c(i) | i < ω be a sequence of non-empty sets and h ∈ ω ω . We define the following cardinal characteristics: v ∀ c,h := min{|F | | F ⊆ c and ¬∃ ϕ ∈ S(c, h) ∀ x ∈ F : x ∈ * ϕ} c ∀ c,h := min{|S| | S ⊆ S(c, h) and ∀ x ∈ c ∃ ϕ ∈ S : x ∈ * ϕ} v ∃ c,h := min{|E| | E ⊆ c and ∀ ϕ ∈ S(c, h) ∃ y ∈ E : y / ∈ ∞ ϕ} c ∃ c,h := min{|R| | R ⊆ S(c, h) and ¬∃ y ∈ c ∀ ϕ ∈ R : y / ∈ ∞ ϕ} The first two types of cardinals are referred to as localisation cardinals, while the latter two are referred to as anti-localisation cardinals. 3 The localisation and anti-localisation cardinals are a generalisation of the cardinals used in Bartoszyński Additionally, when c takes infinitely many infinite values, the localisation and anti-localisation cardinals are already characterised by other well-known cardinals (see [CM19, section 3]), so these cardinals are more interesting when c ∈ ω ω and c > * h. Even then, we have some trivial values: v ∀ c,h is finite and c ∀ c,h = c when 2 The notation x ∈ ∞ y naturally denotes ∃ ∞ i : x(i) ∈ y(i). We originally used / ∈ * instead of / ∈ ∞ , but this turned out to be unnecessarily confusing alongside our use of ∈ * , since / ∈ ∞ is not the negation of ∈ * . 3 A brief note on notation: c ∀ c,h and c ∃ c,h were used in [GS93,Kel08,KS09,KS12,KO14] with the meaning of covering c with slaloms, where the relations for covering are ∈ * and ∈ ∞ , respectively. The notation v ∀ c,h and v ∃ c,h is intended to be read as avoiding or evading such coverings.
h does not go to infinity (Goldstern and Shelah [GS93]); also, c ∃ c,h is finite and v ∃ c,h = c when the quotient h(k) c(k) does not converge to 0 (see [CM19,section 3]). For the other cases, Figure 2 summarises the ZFC-provable inequalities relating localisation and anti-localisation cardinals to other cardinal characteristics. The additional relation between the anti-localisation cardinals and the covering and uniformity numbers of N was previously hinted at for the case of h = 1 in [GS93]; we prove it in general in Lemma 2.3.
Finally, a variation of v ∀ ω,h appears in [BS96, Lemma 2.5] as a characterisation of the linear evasion number e (which coincides with min{b, e}, where e denotes the evasion number).
The ZFC-provable inequalities between localisation and anti-localisation cardinals and other well-known cardinal character- One of the earliest appearance of these cardinals is in Miller's [Mil81] characterisations non(SN ) = min{v ∃ c,h | c ∈ ω ω } (h ≥ * 1) and add(M) = min{b, non(SN )}. 4 Some time later, Goldstern and Shelah [GS93] proved that uncountably many cardinals of the form c ∀ c,h can be pairwise different. Kellner [Kel08] then improved this result by showing the consistency of continuum many pairwise different cardinals of the same type, and Kellner and Shelah [KS09,KS12] included similar consistency results for the type c ∃ c,h ; they also showed that there may be continuum many of both types of cardinals (in the same model). All of these consistency results were proved by using proper ω ω -bounding forcing constructions with normed creatures.
The family of increasing functions is constructed in such a way that for each α ∈ A, v ∃ cα,hα ≤ non(I fα ) ≤ c ∀ aα,dα is provable in ZFC. The poset Q is constructed by a CS (countable support) product of proper ω ω -bounding posets, as in Goldstern and Shelah [GS93], but instead of tree-like posets we use Silver-like posets in the product. These posets are used to add generic reals that increase each v ∃ cα,hα ; in fact, forcing κ α ≤ v ∃ cα,hα is not difficult. On the other hand, guaranteeing that c ∀ aα,dα ≤ κ α holds in the final forcing extension requires careful definition of the increasing functions and strong combinatorics for each forcing in the product. To be more precise, each poset Q dα cα,hα (which is used to increase v ∃ cα,hα ) is obtained by a lim sup creature construction and depends on d α in the sense that this function determines the bigness needed to not increase the cardinals c ∀ a β ,d β for β = α. Standard fusion techniques due to Baumgartner [Bau83,Bau85] are used (in section 3 and section 5) to prove that these posets and their CS products are proper with further properties (such as ω ω -bounding with timely reading of names).
The paper is structured as follows. In section 2, we strengthen the connections Kamo and Osuga found between anti-localisation cardinals and Yorioka ideals' covering and uniformity numbers; besides, we find a (simple) connection between localisation and anti-localisation cardinals. These ensure that, in the Main Theorem, the family of increasing functions satisfies v ∃ cα,hα ≤ non(I α ) ≤ c ∀ aα,dα for every α ∈ A. In section 3, for c, h ∈ ω ω , we define the Silver-like poset we use to increase v ∃ c,h and, by incorporating a norm with sufficiently large bigness, we present sufficient conditions on functions a, e which will guarantee that such a poset does not increase c ∀ a,e . In section 4, we use the results of the previous sections to construct a family of functions as required in the Main Theorem, which is finally proved in section 5. The final section 6 is dedicated to discussions and open questions.

YORIOKA IDEALS AND CARDINAL CHARACTERISTICS
For notational simplicity, we describe the cardinal characteristics we are interested in through relational systems as below.
The dual of R is the relational system R ⊥ := Y, X, a .
Let R := X , Y ,` be another relational system. A pair (F, G) is a Tukey connection from R to R if F : X → X , G : Y → Y and for any x ∈ X and y ∈ Y , F (x)` y implies x`G(y ). When there is a Tukey connection from R to R , we say that R is Tukey-below R , which is denoted by R T R . We say that R and R are Tukey-equivalent, denoted by R ∼ = T R , when R T R and R T R.
In this section, we will use such Tukey connections to prove inequalities between cardinal invariants.
Example 2.2. We give two examples of relational systems.
(1) Let I be a family of subsets of a set X that satisfies (i) [X] <ℵ 0 ⊆ I, (ii) X / ∈ I, and (iii) whenever Y ∈ I and X ⊆ Y , X ∈ I. which are the cardinal invariants associated with I. Note that Cv(I) T I and Cv(I) ⊥ T I, so the well-known fact that add(I) is below both cov(I) and non(I) and that cof(I) is above those three is easily proved through the relational systems.
If J is another family of subsets of X that satisfies (i)-(iii) above and I ⊆ J , then Cv(J ) T Cv(I), so cov(J ) ≤ cov(I) and non(I) ≤ non(J ).
(2) Let c = c(i) | i < ω be a sequence of non-empty sets and h ∈ ω ω . Define If, in addition, c = c (i) | i < ω is a sequence of non-empty sets, h ∈ ω ω and |c(i)| ≤ |c (i)| and h (i) ≤ h(i) for all but finitely many i < ω, then Lc(c, h) T Lc(c , h ) and aLc(c , h ) T aLc(c, h).
Before we engage in the proof, recall that whenever X is an uncountable Polish space and µ is a continuous (i. e. every singleton has measure zero) probability measure on the Borel σ-algebra of X, there is a Borel isomorphism f : X → 2 ω that preserves the measures, i. e. µ(A) is equal to the Lebesgue measure of f [A] for any Borel set A ⊆ X (see e. g. [Kec95,Theorem 17.41]). Therefore, N (X) ∼ = T N and Cv(N (X)) ∼ = T Cv(N ) (where N (X) is the ideal of null sets of X). For the following proof, when c ∈ ω ω , c > 0, and ∃ ∞ i : c(i) ≥ 2, c is the Polish space endowed with the product topology of the discrete spaces c(i) (for i < ω), and µ c denotes the product measure of µ c(i) | i < ω , where µ c(i) is the measure on the power set of c(i) such that each singleton has measure 1 /c(i). It is clear that µ c is a continuous probability measure on the Borel σ-algebra of c.
Proof. To see (a), it is enough to show that Cv(N ( c)) T aLc(c, h) ⊥ . Note that F : c → c, defined as the identity map, and G : S(c, h) → N ( c), defined as G(S) := {x ∈ c | ∃ ∞ i : x(i) ∈ S(i)}, form the corresponding Tukey connection. We only have to check that G(S) ∈ N ( c). Indeed, setting where the last equality follows from the assumption that i<ω h(i) c(i) < ∞. To see (b), first note that, in general, aLc(c, 1) ∼ = T Ed(c) := c, c, = * , where x = * y means ∀ ∞ i : x(i) = y(i). We now first prove the following fact: , and define G : c → d such that for any y ∈ c, G(y)(i) is the unique j ∈ d(i) such that y(i) ∈ b i,j . It is clear that (F, G) is the required Tukey connection.
Thanks to the preceding claim, it suffices to show Cv(N ( d)) T Ed(d). The case d ≤ * 1 is trivial because Ed(d) and, using the fact that 1 + x ≤ e x for every real x, Kamo and Osuga [KO14] proved the following connections between the covering and uniformity numbers of Yorioka ideals and anti-localisation cardinals: (1) Let c ∈ ω ω with c ≥ * 2. If g ∈ ω ω and g(n) ≥ i≤n log 2 c(i) for all but finitely many n < ω, then aLc(c, 1) ⊥ T Cv(J g ). In particular, c ∃ c,1 ≤ cov(J g ) and non(J g ) ≤ v ∃ c,1 .
The two facts above are linked with Yorioka ideals in the sense that J g ⊆ I f ⊆ J f whenever f g (so Cv(J f ) T Cv(I f ) T Cv(J g )). For the purposes of this paper, we improve Kamo and Osuga's results by showing a direct connection to the Yorioka ideals without passing through a family of the form J g .
Lemma 2.4. Let c, h ∈ ω ω , let I n | n < ω be the interval partition such that |I n | = h(n), and let g c,h ∈ ω ω be defined by Proof. It suffices to find F : 2 ω → c and G : S(c, h) → I f such that for any y ∈ 2 ω and S ∈ S(c, h), if ∃ ∞ n : F (y)(n) ∈ S(n), then y ∈ G(S).
For each n < ω, fix a one-to-one map ι n : 2 log 2 c(n) → c(n). For S ∈ S(c, h), enumerate S(n) =: {m S n,k | k ∈ I n } and, whenever k ∈ I n , define σ S (k) := ι −1 n (m S n,k ) when m S n,k ∈ ran ι n ; otherwise σ S (k) is allowed to be anything of length log 2 c(n) .
On the other hand, for y ∈ 2 ω , define F (y)(n) := ι n (y log 2 c(n) ). If F (y)(n) ∈ S(n) for infinitely many n, then for such n there is a k ∈ I n such that F (y)(n) = m S n,k , so σ S (k) = y log 2 c(n) ⊆ y. As there are infinitely many such k, we conclude that y ∈ G(S).
Proof. First note that for any function g , . It suffices to define functions F : b → 2 ω and G : I f → S(b, g) such that for any y ∈ b and X ∈ I f , if F (y) ∈ X, then ∃ ∞ n : y(n) ∈ G(X)(n).
Consider the interval partition I n | n < ω of ω such that |I n | = log 2 b(n) . For each n < ω, fix a one-to-one function ι n : b(n) → 2 In . For y ∈ b, define F (y) as the concatenation of the binary sequences ι n (y(n)) | n < ω . On the other hand, holds for all but finitely many n).
If F (y) ∈ X, then there is an infinite W ⊆ ω such that for any n ∈ W , σ X (k) ⊆ F (y) for some k ∈ J n ; so σ X (k) In = F (y) In = ι n (y(n)) and hence we have y(n) ∈ G(X)(n).
The following result shows a connection between localisation cardinals and antilocalisation cardinals. This is useful to include localisation cardinals of the type c ∀ in our main result.
Lemma 2.6. Let c, h ∈ ω ω with c > 0 and h ≥ * 1. If c is a function with domain ω and h ∈ ω ω such that for all but finitely many i < ω, Proof. It suffices to show the result for the case when The next lemma is more or less the converse of the preceding one. We prove it only for completeness' sake, and it will not be used in this paper.

Proof. It suffices to show the result for the case when
. This means that for all but finitely many i < ω, By Lemma 2.6 and Lemma 2.7 applied to c = c, h = 1 and h = c − 1, it follows that aLc(c, 1) ∼ = T Lc(c, c − 1) (although it is also quite simple to prove this directly). More properties of localisation and anti-localisation cardinals can be found in e. g. [GS93,CM19].
The following lemma shows how functions should be related to get a particular chain of inequalities between anti-localisation cardinals, localisation cardinals and uniformity numbers of Yorioka ideals, which will be useful for the main result of this paper. Then Proof. This is a direct consequence of Lemma 2.4, Lemma 2.5 and Lemma 2.6.

MOTIVATIONAL EXAMPLE
We will now define a creature forcing poset (cf. [RS99] for the original seminal work on this technique) whose countable support product we will use to increase the value of v ∃ c,h to be at least some prescribed cardinal κ. Through careful choice of the function parameters (and some additional auxiliary functions), we will show in section 5 that v ∃ c,h is indeed forced to be exactly κ, and then use a countable support product of many such creature forcing posets with varying function parameters to force uncountably many Yorioka ideals' uniformity numbers to be different.
We define a forcing poset Q c,h as follows: A condition p ∈ Q c,h is a sequence of creatures p(n) such that each p(n) is a non-empty subset of [c(n)] ≤h(n) and such that, letting the norm · c,h,n be defined by Figure 3 for an illustration of the initial segment of a condition in this forcing.) Given a condition p such as above, the finite initial segments in p k+1 (for k < ω) are sometimes referred to as possibilities and denoted by poss(p, ≤k) : We may also use the notation poss(p, <k) := poss(p, ≤k − 1). When η ∈ poss(p, ≤k), we write p ∧ η to denote η p [k+1,ω) . 6 We denote the indices of the non-trivial creatures by split(p) := k < ω | |p(k)| > 1 and, for n < ω, denote the n-th member of split(p) by s n (p). For p, q ∈ Q c,h , define q ≤ n p as "q ≤ p and q sn(q)+1 = p sn(q)+1 ", which means that q is stronger than p and they are identical up to (including) the n-th non-trivial creature.
Observation 3.2. In this section, we often work with the set Q * c,h of conditions p ∈ Q c,h such that p(s n (p)) c,h,sn(p) ≥ n + 1 (i. e. the n-th non-trivial creature has norm at least n + 1). It is clear that Lemma 3.3. The poset Q c,h adds a generic slalomṠ ∈ S(c, h) such that Q c,h forces that any ground-model real in c is caught infinitely often byṠ.
as the unique member of p∈Ġ p(n) (whereĠ is the generic filter), which clearly is a name for a member of [c(n)] ≤h(n) . Now assume that x ∈ V ∩ c and fix a condition p ∈ Q c,h and an n 0 < ω. Pick some k ≥ n 0 such that p(k) c,h,k ≥ 1 and strengthen p to q by setting q(k) = {t} p . . . for some t ∈ p(k) that contains x(k) (which exists because, by the definition of the norm, M c,h,n ≥ 1 iff M = c(n)). Hence q ≤ p and q x(k) ∈Ṡ(k).
We now show that the poset Q c,h is indeed a proper ω ω -bounding poset and that, moreover, Q * c,h satisfies strong axiom A. It is clear that |Q c,h | = c so, assuming CH, this poset has ℵ 2 -cc and hence preserves cardinalities and cofinalities.
Proof. Enumerate poss(p, ≤s n (p)) =: is precisely the condition we are looking for.
Lemma 3.5. The poset Q * c,h satisfies strong axiom A. Concretely, it satisfies: (a) For any p, q ∈ Q * c,h and m ≤ n < ω, if p ≤ n q, then p ≤ m q and p ≤ q. (b) Whenever p n | n < ω is a sequence in Q * c,h which satisfies p n+1 ≤ n p n for every n < ω, there is some q ∈ Q * c,h such that q ≤ n p n for every n < ω.
(c) If A ⊆ Q * c,h is an antichain, p ∈ Q * c,h and n < ω, then there is a condition q ≤ n p in Q * c,h such that only finitely many r ∈ A are compatible with q. In particular, Q c,h is proper and ω ω -bounding.
Proof. Property (a) follows immediately from the definition. To see property (b), By the definition of q, q ≤ n p n . We now show property (c). Set D as the set of conditions p ∈ Q c,h which are either stronger than some member of A or incompatible with every member of A. It is clear that D is an open dense set, so we can find q ≤ n p in Q * c,h as in Lemma 3.4, that is, such that q ∧ η ∈ D for any η ∈ poss(q, ≤s n (q)), which means that there is at most one r η ∈ A weaker than q ∧ η. Hence, if r ∈ A is compatible with q, then it must be compatible with q ∧ η for some η ∈ poss(q, ≤s n (q)) (of which there are only finitely many), so r = r η because A is an antichain.
A sequence p n | n < ω as in property (b) is usually known as a fusion sequence, and the q obtained in its proof is known as the fusion of p n | n < ω . As an application of this notion, we prove the properties of continuous and timely reading of names and few possibilities for the forcing Q c,h . Definition 3.6. Let p ∈ Q c,h and letτ be a Q c,h -name for a function from ω into the ground model V .
(1) We say that p readsτ continuously iff for each n < ω there is some k n such that for each η ∈ poss(p, ≤k n ), p ∧ η decidesτ n .
(2) We say that p readsτ timely iff for each n ∈ split(p) and each η ∈ poss(p, ≤n), p ∧ η decidesτ n . (This is equivalent to continuous reading with k n = n for each n ∈ split(p).) The term "continuous reading" refers to the fact that, in a sense, there is a continuous function from "branches" in p to V ω , coded by a function from initial segments of p to initial segments ofτ . It is clear that timely reading of a name implies continuous reading thereof; we will only be using the stronger property of timely reading in the rest of this paper.
Lemma 3.7 (timely reading of names). Let p ∈ Q c,h and letτ be a Q c,h -name for a function from ω into the ground model V . Then there is a condition q ≤ p in Q c,h such that q readsτ timely.
Proof. Denote by D n the set of conditions in Q c,h which decide τ n , which is an open dense subset of Q c,h . By recursion on < ω, define an increasing function f ∈ ω ω and a sequence p | < ω of members of Q * c,h such that For the construction, choose any p 0 ≤ p in Q * c,h and let f (0) := s 0 (p 0 ), which clearly satisfy (i) and (ii). Now assume that p and f ( ) have been constructed and that they satisfy (ii). Find p +1 ∈ Q * c,h by application of Lemma 3.4 to , p and D f ( ) . As in Lemma 3.5 (b), define q = q(k) | k < ω such that q (f ( )−1,f ( )] := p (f ( −1),f ( )] for any < ω (letting f (−1) := −1). It is clear that s (q) = f ( ), even more, q(f ( )) c,h,f ( ) = p (f ( )) c,h,f ( ) ≥ + 1, so q ∈ Q * c,h . Besides, q ≤ p for any < ω. On the other hand, by (iv), q ∧ η ∈ D f ( ) for any η ∈ poss(q, ≤f ( )).
Lemma 3.8 (few possibilities). Let g ∈ ω ω be a function going to infinity. If p ∈ Q c,h , then there is a condition q ≤ p in Q * c,h such that | poss(q, <s n (q))| < g(s n (q)) for all n < ω.
As in previous arguments, define the fusion q ∈ Q * c,h of the sequence p n | n < ω by q(k) := p n (k) whenever k ∈ (f (n − 1), f (n)]. It is easy to see that q is as required.
We are now interested to know for which functions b, g ∈ ω ω the poset Q c,h will not increase the cardinal v ∃ b,g (i. e. any slalom in S(b, g) from the generic extension anti-localises some real in b from the ground model) or the cardinal c ∀ b,g (i. e. any real in b from the generic extension is localised by some slalom in S(b, g) from the ground model). This will be the key point to understand the relation between the functions so that the Main Theorem can be proved using a countable support product of our posets increasing cardinals of the form v ∃ c,h . To ensure such preservation, it seems that our forcing requires the norms on the creatures to satisfy bigness (in the sense of [FGKS17]    Proof. Partition m into sets {a j | j < } of size at most k with ≤ d(n). Define Thanks to bigness, timely reading of names can be strengthened in the following way: Lemma 3.12 (early reading). Let A n | n < ω be a sequence of non-empty finite sets, and letτ be a Q d c,h -name for a member of n<ω A n . If d goes to infinity and i<n A i ≤ d(n) for all n < ω, then for any p ∈ Q d c,h , there is some q ≤ p which readsτ early, that is, for any n < ω and η ∈ poss(q, <n), q ∧ η already decidesτ n .
Proof. Without loss of generality, by Lemma 3.7 we may assume that p ∈ Q * d c,h readsτ timely, that is, p ∧ η decidesτ n for any η ∈ poss(p, ≤n) and n ∈ split(p); by Lemma 3.8 we may also assume that | poss(p, <n)| < d(n) for any n ∈ split(p).
We construct q(k) by recursion on k < ω. When k / ∈ split(p), let q(k) := p(k). Now assume that k ∈ split(p) and work with p k := q k p [k,ω) . Enumerate poss(p k , <k) =: {η j | j < m}, where m < d(k). By recursion on j ≤ m, define M j ⊆ p(k) such that Assume we already have M j . Define the function h j : Hence, by Lemma 3.10, there is some h,k − 1 and, by property (iv), p k ∧ (η j {t}) forcesτ k = r j for any t ∈ q(k) and j < m, which means that q k+1 p [k+1,ω) ∧ η j forcesτ k = r j .
By the construction, when k ∈ split(p), q(k) d c,h,k ≥ p(k) d c,h,k − 1, so q ∈ Q d c,h . If k ∈ split(p) and η ∈ poss(q, <k), then q ∧ η decidesτ k . Now, if k ∈ ω split(p) and η ∈ poss(q, <k), then there is a unique η ∈ poss(q, < k ) extending η, where k is the smallest member of split(p) above k, so q ∧ η = q ∧ η . As this decideṡ τ k , it is clear that it also decidesτ k .
The following result gives sufficient conditions on functions a, e to guarantee that Q d c,h does not increase c ∀ a,e . Hereafter, we fix the notation c h (k) := |[c(k)] ≤h(k) |.
Lemma 3.13. Assume that c, d, h ∈ ω ω are as in Definition 3.9 with d going to infinity, a, e ∈ ω ω with a > 0 and e going to infinity, and that they satisfy (L1) k<n a(k) ≤ d(n) and k<n c h (k) ≤ e(n) for all but finitely many n, and (L2) lim k→∞ min c h (k) e(k) , a(k) d(k) = 0. Then Q d c,h forces that any real in a is localised by some member of S(a, e) ∩ V .
Proof. We show that, whenever p ∈ Q d c,h andẋ is a Q d c,h -name for a real in a, then there is some ϕ ∈ S(a, e) ∩ V and some q ≤ p in Q d c,h forcingẋ(k) ∈ ϕ(k) for all but finitely many k. Without loss of generality, we assume that (i) p ∈ Q * d c,h readsẋ early (by (L1) and Lemma 3.12), (ii) for any k ∈ split(p), | poss(p, <k)| ≤ d(k) and min c h (k) e(k) , a(k) d(k) ≤ 1 2 | poss(p, <k)| (by (L2) and Lemma 3.8), and (iii) | poss(p, <k)| ≤ e(k) for every k < ω (by (L1)).
By recursion on k, we construct q(k) and ϕ(k) according to the following case distinction: First, let p k := q k p [k,ω) .

Now consider the case when
We show how to find q(k) ⊆ p(k) with q(k) Assume we already have M j . By (i), p k ∧ (η j {t}) decidesẋ(k) for every t ∈ M j , which is a value in a(k). At the end of the construction, it is clear that q forcesẋ(k) ∈ ϕ(k) for any sufficiently large k, and that ϕ(k) ∈ [a(k)] ≤e(k) .
Observation 3.14. The function c h is used as an upper bound of |p(k)| | k < ω for any p ∈ Q d c,h , since p(k) ⊆ [c(k)] ≤h(k) . In the same way, we could use c h (pointwise exponentiation) instead (as long as h ≥ * 2), because |[m] ≤k | ≤ m k whenever m, k < ω and k = 1 (since |[m] ≤1 | = m + 1). In fact, in the following sections, every instance of c h could be replaced by c h without affecting the results and the proofs.
Thanks to the Tukey connection constructed in Lemma 2.6, we can now easily deduce sufficient conditions on functions b, g to guarantee that v ∃ b,g is not increased by Q d c,h . Corollary 3.15. Assume that c, d, h ∈ ω ω are as in Definition 3.9 with d going to infinity, b, g, d ∈ ω ω with b, g > 0, e(k) := b(k) g(k) − 1 going to infinity, and that they satisfy (AL1) k<n b g (k) ≤ d(n) and k<n c h (k) ≤ e(n) for all but finitely many n, and Then Q d c,h forces that any slalom in S(b, g) is anti-localised by some member of b ∩ V .
Proof. Set a(k) := b g (k). By Lemma 2.6, there is a definable Tukey connection (F, G) which witnesses aLc(b, g) T Lc(a, e) (even in forcing-generic extensions). As a and e satisfy the assumptions in Lemma 3.13, in any Q d c,h -generic extension, any real in a is localised by some slalom in S(a, e) ∩ V . Hence any slalom in S(b, g) is anti-localised by some member of b ∩ V .

LOTS AND LOTS OF AUXILIARY FUNCTIONS
To show that we can separate uncountably many different Yorioka ideals' uniformity numbers, we could in principle define two sequences of integers acting as universal bounds on our function parameters and auxiliary functions (similar to [GS93]). However, for the sake of clarity we will give the definition of the sequences as part of an inductive definition in the construction of the auxiliary functions. We stress that there also is an a priori definition, which would, however, make the text less readable.
Either way, we fix two sequences n − k , n + k of natural numbers ≥ 2 such that (i) n − k · n + k < n − k+1 for any k < ω, and (ii) lim n→∞ log n − k n + k = ∞. Given Lemma 3.13 and Corollary 3.15 above, we now make the following definition.
9 We can simplify property (S5) by restricting it to = 1: Assume the statement holds for some > 1. Let f α = f α • pow . Then f bα,gα ≤ * f α and f α g cα,hα still holds (by the definition of ). This is why we can work with f α instead of f α , in which case property (S5) is already satisfied for = 1 and property (S6) remains true for f α . Properties (S3)-(S6) ensure that, by Lemma 2.8, v ∃ cα,hα ≤ non(I fα ) ≤ v ∃ bα,gα ≤ c ∀ aα,dα . When we prove the Main Theorem in the following section, we will aim to force all these cardinals to be equal to some predetermined κ α for each α. We therefore increase the cardinals v ∃ cα,hα by using a countable support product of posets of the form Q dα cα,hα , while at the same time ensuring that c ∀ aα,dα does not exceed the desired value. Property (S2) guarantees that each poset Q dα cα,hα is nonempty and, thanks to Lemma 3.13, property (S7) will ensure that Q d β c β ,h β will not increase c ∀ aα,dα for β = α. To this end, it is also necessary that <k a α ( ) ≤ d β (k) for all but finitely many k (property (L1) in Lemma 3.13), which is a consequence of property (S1) because <k a α ( ) ≤ <k n + < n − k ≤ d β (k) (the second inequality follows from property (i) above Definition 4.1 by induction; note that the case α = β is also allowed). In fact, property (S1) is what allows us to explicitly control the functions and the number of possibilities of the creatures in the poset.
Theorem 4.2. There are bounding sequences n − k , n + k such that there is an un- Proof. For clarity, we first explain how to construct a suitable family for one single α, that is, a suitable tuple of the form (a, d, b, g, f, c, h) (as in Definition 4.1 for the case |A| = 1, for which (S7) is vacuous). For motivational purposes, assume we have already defined n − k and d(k) ≥ n − k , and the values of all the functions at < k (with the exception of f , which depends on some interval partition as indicated in the following argument). From here, the order in which the values should be defined is h(k), g(k), b(k), f I k (where I k depends on the values of g up to k), c(k), a(k) and n + k . For k = 0, n − 0 and d(0) can be chosen arbitrarily such that 2 < n − 0 < d(0). To have property (S2), it suffices to define h(k) such that 1 d(k) log d(k) (h(k) + 1) ≥ k + 1, so we let h(k) := d(k) (k+1)·d(k) . (Later, when we define c(k), it will be clear that c(k) > h(k).) The value of g(k) is determined so that f (k) (which will be defined a few steps later) can satisfy property (S6), but for now we will just assume that we have already defined it and will later explain how to define it according to our needs. (The value of g(k) will only depend on h(k) as well as h( ) and g( ) for < k.) Now, to fulfil (S3), we let b(k) := 2 g(k)+d(k) (such that log 2 b(k) will be an integer, making the definition of f b,g a bit nicer). Now recall that f b,g is defined, along the interval partition I n | n < ω of ω which satisfies |I n | = g(n), as f b,g (j) = ≤n log 2 b( ) for any j ∈ I n . In our construction, we may assume that we have already defined g( ) and I for any ≤ k, so we explain how to define f I k such that property (S5) is satisfied -even more, such that f b,g (j) ≤ f (j) for any j ∈ I k . We also need to make sure that f is increasing, so we define for any j ∈ I k . This definition allows f to be increasing when attached to the previous intervals because Recall that g c,h is defined, along the interval partition J n | n < ω of ω which satisfies |J n | = h(n), as g c,h (j) = log 2 c(n) for all j ∈ J n . In order to have property (S6), it suffices to define c(k) such that f (j k+2 ) ≤ log 2 c(k) for any j ∈ J k . If we can ensure that j k+2 ∈ I k for any j ∈ J k , then Now, to ensure that j k+2 ∈ I k for any j ∈ J k , it suffices to have that min(J ) +1 = min(I ) for any ≤ k + 1 (as min(J 0 ) = 0 and min(J +1 ) = min(J ) + h( ) as well as min(I +1 ) = min(I ) + g( ), the values of min(J k+1 ) and min(I k+1 ) are already known). The choice of g(k), which we had postponed until now, will guarantee this fact. So we assume that we had already ensured min(J k ) k+1 = min(I k ) from the beginning (so, before n − k is even determined; also note that this is true for k = 0) and, after defining h(k), we let g(k) := min(J k+1 ) k+2 − min(I k ). This implies that min(I k+1 ) = min(I k ) + g(k) = min(J k+1 ) k+2 as required.
We can choose a(k) above c h (k) and b g (k), and define n + k := a(k), which guarantees property (S4). Note that by our definitions, we have which together ensure property (S1).
We can then choose n − k+1 above n − k · n + k (note that log n − k n + k ≥ log d(k) (h(k) + 1) > k) and define d(k + 1) to be larger than n − k+1 , and then the iterative construction continues for k + 1. Now, we define both the bounding sequences and the suitable family F of size continuum inductively. More or less the same strategy as above will suffice to define the values at a fixed k, but some extra work is needed in order to guarantee property (S7). We construct tuples of functions (a t , d t , b t , g t , f t , c t , h t ) | t ∈ 2 <ω such that (i) |a t | = |d t | = |b t | = |g t | = |c t | = |h t | = |t| and |f t | = k<|t| g t (k), and (ii) if t ⊆ t in 2 <ω , then a t ⊆ a t , and likewise for the other functions.
Denote by ¡ the lexicographic order in both 2 ω and 2 n for any n < ω. We construct (a t , d t , b t , g t , f t , c t , h t ) | t ∈ 2 n by recursion on n. When n = 0, it is clear that a , d , etc. are defined as the empty sequence. Assuming we have reached step n of the construction, we show how to obtain the functions for t ∈ 2 n+1 . We do this in the following two steps: 1. Assuming we have already defined d t (n), we define h t (n), g t (n) and so on in the same way as in the definitions for the simple case |A| = 1. 2. By recursion on 2 n+1 , ¡ , we construct d t (n) for t ∈ 2 n+1 . We first show step 1. At this point, we have sequences I t +1 | < n and J t +1 | < n of consecutive intervals covering an initial segment of ω such that |I t +1 | = g t ( ) and |J t +1 | = h t ( ). In fact, I t +1 = [|f t |, |f t | + g t ( )) for any < n. Note that we can already define min(I t n ) and min(J t n ) at this stage (considering that we are constructing interval partitions of ω). Further assume that min(I t n ) = min(J t n ) n+1 , which is trivially true for n = 0.
, min(J t n ) + h t (n)), and b t (n) := 2 gt(n)+dt(n) . Let I t n := [|f t n |, |f t n | + g t (n)) and, for k ∈ I t n , define By definition, |f t | = |f t n | + g t (n) = max(I t n ) + 1. Finally, let and choose a t (n) above c ht t (n) and b gt t (n). As in the case of |A| = 1, d t (n) < h t (n) < g t (n) < b t (n) < c t (n) < a t (n) (it will be clear from the next step that d t will always take values ≥ 2) and the other three functions to be bounded lie somewhere in between.
To see step 2, start by choosing d0 n+1 (n) > d0 n (n − 1) · a1 n (n − 1) + 2 (in case n = 0, just choose d0 1 (0) to be any number above 2). Now assume that we have d t (n) and that t + ∈ 2 n+1 is the immediate successor of t with respect to ¡. Let d t + (n) := (n + 1) · a t (n). This completes the construction. Note that, whenever t, t ∈ 2 n+1 and t ¡ t , then at(n) d t (n) ≤ 1 n+1 , which is what ultimately guarantees property (S7). Now we finally show that F = (a α , d α , b α , g α , f α , c α , h α ) | α ∈ 2 ω is a suitable family for the bounding sequences n − k and n + k , that is, it satisfies the properties in Definition 4.1. Properties (S1)-(S6) are immediate by construction, as in the case |A| = 1. Note that I α n := I α n+1 n and J α n := J α n+1 n (for n < ω) define interval partitions of ω such that |I α n | = g α (n) and |J α n | = h α (n), so properties (S5) and (S6) can be proved in the same way as before.

THE GRAND FINALE
Now that we have defined suitable families, we can put everything together to prove the Main Theorem: Theorem 5.1. Assume CH and let κ α | α ∈ A be a sequence of infinite cardinals such that |A| ≤ ℵ 1 and κ ω α = κ α for every α ∈ A. Given bounding sequences n − k , n + k and a suitable family (Recall Definition 4.1 for the definition of a suitable family and Definition 3.9 for the definition of the poset Q dα cα,hα .) This section is dedicated to proving the theorem above. Fix a disjoint family K α | α ∈ A such that |K α | = κ α and let K := α∈A K α . Hence, we can express Q as the countable support product of Q ξ | ξ ∈ K , where Q ξ := Q dα cα,hα whenever ξ ∈ K α . For any p ∈ Q, ξ ∈ supp(p) and k < ω we write p(ξ, k) := p(ξ)(k).
By the following lemma, without loss of generality we may assume that any p ∈ Q is modest (unless we explicitly say the opposite), that is, for every < ω there is at most one ξ ∈ supp(p) such that ∈ split(p(ξ)).

Lemma 5.2 (modesty). The set of modest conditions in Q is dense.
Proof. Fix p ∈ Q. By induction on n < ω it is possible to define a function f = (f 0 , f 1 ) : ω → supp(p) × ω such that (i) f 1 is strictly increasing, (ii) f 1 (n) ∈ split(p(f 0 (n))) and the norm of p(f 0 (n)) (with respect to Q f 0 (n) ) is larger than n + 1, and (iii) for any ξ ∈ supp(p), the set f −1 0 [{ξ}] is infinite. Define q with supp(q) = supp(p) such that q(ξ, k) = p(ξ, k) whenever (ξ, k) ∈ ran f , otherwise q(ξ, k) is a singleton contained in p(ξ, k). By the definition of f it is clear that q ∈ Q and q ≤ p. Moreover, by (i), q is modest.
Thanks to modesty, Q behaves very much like a single forcing as in section 3. Before we show Theorem 5.1, we revisit the notation and results of section 3 and revise them for the context of Q. The proofs of properness, ω ω -bounding and timely reading (as well as the related lemmata) are quite similar to the proofs for the single forcing. Notation 5.3. In the context of Q, for any p ∈ Q and k, n < ω: (1) split(p) := ξ∈supp(ξ) split(p ξ ), which is a disjoint union by modesty.
(2) s n (p) denotes the n-th member of split(p).
Lemma 5.4. If n < ω, p ∈ Q, and D ⊆ Q is open dense, then there is some q ≤ n p in Q such that for any η ∈ poss(q, ≤s n (q)), q ∧ η ∈ D.
Define Q * as the set of conditions in p ∈ Q such that for any n < ω, the norm of p(ξ p (s n (p)), s n (p)) is above n + 1. Note that the condition q found in Lemma 5.2 is actually in Q * , so this set is dense in Q. Also, if p ∈ Q * , α ∈ A, and ξ ∈ K α ∩ supp(p), then p(ξ) ∈ Q * ξ := Q * dα cα,hα . Lemma 5.5 (fusion). Let p n , F n | n < ω be a sequence such that (i) p n ∈ Q * and F n ⊆ K is non-empty finite, (ii) p n+1 ≤ n,Fn p n , (iii) F n ⊆ F n+1 and W := n<ω F n is equal to n<ω supp(p n ).
Then there is a condition q ∈ Q with supp(q) = W such that q ≤ n,Fn p for every n < ω.
Proof. For each ξ ∈ W , let n ξ := min{n < ω | ξ ∈ F n }. It is clear that p n (ξ) | n ≥ n ξ is a sequence in Q * ξ and that p n+1 (ξ) ≤ n p n (ξ). Thus, there is a fusion q(ξ) ∈ Q * ξ of that sequence as in the proof of Lemma 3.5. Note that s k (q(ξ)) = s k (p n (ξ)) for any n ≥ n ξ and k ≤ n.
Lemma 5.6. Let χ be a sufficiently large regular cardinal and let N H χ be countable such that Q ∈ N . Let D n | n < ω be a sequence of open dense subsets of Q such that each D n ∈ N . If p ∈ Q ∩ N , then there is a condition q ≤ p in Q such that for any n < ω and η ∈ poss(q, s n (q)), q ∧ η is stronger than some member of D n ∩ N .
Each step of this construction takes place inside N , though it is very likely that the final sequence is outside N . Figure 4 illustrates the idea of the following construction. Choose some p ≤ p in Q * such that ξ 0 ∈ supp(p ) and s 0 (p ) = s 0 (p (ξ 0 )), and find p 0 ≤ 0 p by application of Lemma 5.4 to D 0 . Assume that we have constructed p n . Let m n := |L n | − 1 = (n + 1) 2 − 1. By (iv), p n satisfy the lemma for the splitting levels k ≤ m n . The idea in the construction of p n+1 is as follows: (1) We guarantee that s k (p n+1 ) = s k (p n ) for each k ≤ m n .
(2) For each k ≤ n, fix a sufficiently high splitting level of p n (ξ k ), which will be s mn+k+1 (p n+1 ), and use Lemma 5.4 to guarantee that (iv) holds at the splitting level m n + k + 1.
(3) Add ξ n+1 to the support of the condition constructed so far, extend the ξ n+1 -coordinate so that its first splitting level is sufficiently high (this will be s mn+n+2 (p n+1 )), and use Lemma 5.4 to guarantee that (iv) is satisfied at this level. (4) At ξ n+1 , by a recursion of n+1 steps choose a sufficiently high splitting level and ensure that (iv) will be satisfied at this level by applying Lemma 5.4. The last splitting level will be s mn+2(n+1)+1 .
We give a formal specification of the construction (with corresponding enumeration). By recursion on k < 2(n + 2), construct p n,k as follows: (1) Let p n,0 := p n .
(2) Having defined p n,k for k ≤ n, choose an ∈ split(p n,k (ξ k )) larger than s mn+k (p n,k ), choose p n,k ≤ mn+k p n,k such that p n,k (ξ k , ) = p n,k (ξ k , ) and = s mn+k+1 (p n,k ), and find p n,k+1 ≤ mn+k+1 p n,k by application of Lemma 5.4 to D mn+k+1 .
By Lemma 5.5, we can find a fusion q of p n | n < ω with supp(q) = K ∩ N such that q ≤ n,Fn p n for any n < ω. By (iii), q ≤ |Ln|−1 p n for each n < ω, so (iv) implies that q is as desired.
Proof. Properness is immediate by applying Lemma 5.6 to the enumeration of all the open dense subsets in N .
To show that Q is ω ω -bounding, letẋ be a Q-name for a member of ω ω and let p ∈ Q. Let χ be a sufficiently large regular cardinal and let N H χ be countable such that Q, p,ẋ ∈ N . Let D n be the set of conditions of Q that decideẋ(n), which is an open dense set that belongs to N . Find q ≤ p by application of Lemma 5.6 to D n | n < ω , which implies that q ẋ ∈ B n , where B n := {k < ω | ∃ η ∈ poss(q, s n (q)) : q ∧ η ẋ(n) = k} is a finite set. Hence, q forces thatẋ is bounded by f ∈ V , where f (n) := max(B n ).
Corollary 5.8 (timely reading of names). Let p ∈ Q and letτ be a Q-name for a function from ω into the ground model V . Then there is a condition q ≤ p in Q such that q readsτ timely.
Proof. A fusion argument as in the proof of Lemma 5.6 works, but here the dense sets need to be defined within the construction of the fusion sequence. Concretely, start with a sufficiently large regular cardinal χ and a countable model N H χ such that Q, p,τ ∈ N , and construct the fusion sequence p n | n < ω satisfying conditions (i)-(iii) and (v) in the proof of Lemma 5.6 -but instead of (iv) demand (iv') for any k < |L n | and η ∈ poss(p n , s k (p n )), p n ∧ η ∈ D k , where D k is the (open dense) set of conditions in Q * that decideτ s k (pn) .
The construction is the same as in Lemma 5.6 except that the dense sets D k are defined just before each application of Lemma 5.4.
Proof. This follows by a typical ∆-system argument under CH.
Define q as in Lemma 5.5.
Lemma 5.11 (early reading). Let X k | k < ω be a sequence of non-empty sets with |X k | ≤ n + k , and letτ be a Q-name for a member of k<ω X k . Then for any p ∈ Q, there is a condition q ≤ p in Q which readsτ early, that is, for any n < ω and η ∈ poss(q, <n), q ∧ η already decidesτ n .
The lower bound. Fix some α ∈ A. We prove Q v ∃ cα,hα ≥ κ α when κ α > ℵ 1 (otherwise it is trivial) as follows (analogously to the proof of Lemma 3.3): LetḞ be a Q-name for a subset of c α of size < κ α . Without loss of generality, we may assume that there is some ν < κ α such that Q forces |Ḟ | ≤ ν. 10 By ℵ 2 -cc, there is some B ∈ V with |B| ≤ max{ℵ 1 , ν} < κ α such thatḞ is a Q B -name. Hence there is some ξ ∈ K α B.
It suffices to show that for any Q B -nameẋ for a member of c α , Q forces thaṫ x(k) ∈Ṡ ξ (k) for infinitely many k < ω. Only for this argument, we can briefly forget about modesty. Fix p ∈ Q and some n 0 < ω and, without loss of generality, 10 Concretely, there are p 0 ∈ Q and ν < κ α such that p 0 |Ḟ | = ν. So we just replaceḞ by some Q-nameḞ for a subset of c α of size ≤ ν such that p 0 Ḟ =Ḟ . also assume ξ ∈ supp(p). Pick a k ≥ n 0 such that the norm of p(ξ, k) is at least 1; afterwards, find an r ≤ p B which decidesẋ(k) = z. Define p ∈ Q such that supp(p ) = supp(r) ∪ supp(p), p (γ) := r(γ) for γ ∈ B and p (γ) := p(γ) otherwise. Finally, strengthen p to q with the same support by first setting q K {ξ} := p K {ξ} , and then setting q(ξ, k) := {t} for some t ∈ p (ξ, k) which contains z (which exists by the definition of the norm, as explained in Lemma 3.3) and q(ξ, n) := p (ξ, n) when n = k. Hence q ≤ p and q ẋ(k) ∈Ṡ ξ (k).
This argument shows that given any family of size less than κ α , this set cannot be anti-localising in the sense of v ∃ cα,hα , and hence Q v ∃ cα,hα ≥ κ α . The upper bound. This argument is very similar to Lemma 3.13. Fix α ∈ A and let C α := {K β | κ β ≤ κ α , β ∈ A}. Note that |C α | = κ α and Q C α forces that c = κ α . To show Q c ∀ aα,dα ≤ κ α it suffices to prove that for any p ∈ Q and any Q-nameẋ for a member of a α , there is some q ≤ p in Q and some Q C αnameφ such that q "φ ∈ S(a α , d α ) andẋ ∈ * φ ". (This means that whenever G is Q-generic over V , any member of a α is localised in V [G] by some slalom in S(a α , d α ) ∩ V [G ∩ Q C α ], where the latter set has size κ α .) Without loss of generality, we may assume that (i) p ∈ Q * readsẋ early (by Lemma 5.11 because a α (k) ≤ n + k ), and (ii) for any ξ ∈ supp(p) C α , if β ∈ A and ξ ∈ K β , then for any k ∈ split(p(ξ)), when ξ is as above, or the identity on ω otherwise).
In the end, both q and ϕ are as required.

OPEN QUESTIONS
Concerning the consistency of infinitely many pairwise different cardinals associated with Yorioka ideals, the following summarises the current open questions. It is also interesting to consider the consistency of the conjunction of the statements above.
As mentioned in the introduction, Kamo and Osuga [KO14] used the existence of a weakly inaccessible cardinal to force the statement in (Q1), but its consistency is still unknown assuming only ZFC. We believe that extending the ideas in our construction with lim inf techniques as in [KS12] would work to solve this problem (even simultaneously with (Q2)).
Very little is known about the additivity and cofinality of Yorioka ideals. Even the following question is still open (see also [CM19, section 6]).

Question D.
Is it consistent with ZFC that there are two Yorioka ideals with different additivity numbers (or cofinality numbers)?
Any idea to solve this question in the positive could be used to prove the consistency of (Q4) using a lim sup creature construction as in this paper. However, as the additivity numbers of Yorioka ideals are below b, the typical ω ω -bounding creature forcing notions will not work to prove the consistency of (Q3).
As for the localisation and anti-localisation cardinals, the remaining open problems are the following.