Preservation of splitting families and cardinal characteristics of the continuum

We show how to construct, via forcing, splitting families that are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń’s diagram, m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{m}$$\end{document} (2-Knaster), p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{p}$$\end{document}, h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{h}$$\end{document}, the splitting number s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{s}$$\end{document} and the reaping number r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{r}$$\end{document}.


Introduction
In this paper we present a method to preserve certain splitting families along finite support iterations.These splitting families are constructed via forcing, using specific uncountable 2-edge-labeled graphs 1 as support.The main application of this method is a forcing model where many classical cardinal characteristics of the continuum are pairwise different, including the splitting number s and the reaping number r.
We assume that the reader is familiar with Cichoń's diagram (Figure 1) containing the characteristics that we will call Cichoń-characteristics.We also investigate some of the characteristics in the Blass diagram [Bla10,Pg. 481].Figure 2 illustrates both diagrams combined, along with all the ZFC-provable inequalities that we are aware of.See [Bla10,BJ95] for the definitions and the proofs for the inequalities (with the exception of cof(M) ≤ i, which was proved in [BHHH04]).In the following, we only give the definitions of the non-Cichoń-characteristics that we will investigate in this paper.In the version on the right, the two "dependent" values add(M) = min{b, cov(M)} and cof(M) = max{non(M), d} are removed; the "independent" ones remain (nine entries excluding ℵ 1 , or ten including it).An arrow x → y means that ZFC proves x ≤ y.
(ii) if ⟨I n : n < ω⟩ is an interval partition of ω then n∈a I n ∈ D for some a ∈ [ω] ℵ0 .The groupwise density number g is the smallest size of a collection of groupwise dense sets whose intersection is empty.(6) The distributivity number h is the smallest size of a collection of dense subsets of ⟨[ω] ℵ0 , ⊆ * ⟩ whose intersection is empty.(7) Say that a ∈ [ω] ℵ0 is a pseudo-intersection of F ⊆ [ω] ℵ0 if a ⊆ * b for all b ∈ F .(8) The pseudo-intersection number p is the smallest size of a filter base of subsets of [ω] ℵ0 without pseudo-intersection.(9) The tower number t is the smallest length of a (transfinite) ⊆ * -decreasing sequence in [ω] ℵ0 without pseudo-intersection.(10) Given a class P of forcing notions, m(P) denotes the minimal cardinal κ such that, for some Q ∈ P, there is some collection D of size κ of dense subsets of Q without a filter in Q intersecting every member of D. (11) Let P be a poset.A set A ⊆ P is k-linked (in P) if every k-element subset of A has a lower bound in P. A is centered if it is k-linked for all k ∈ ω. (12) A poset P is k-Knaster, if for each uncountable A ⊆ P there is a k-linked uncountable B ⊆ A. And P has precaliber ℵ 1 , if such a B can be chosen centered.For notational convenience, 1-Knaster means ccc, and ω-Knaster means precaliber ℵ 1 .(13) For 1 ≤ k ≤ ω denote m k := m(k-Knaster) and m := m 1 .We also set m 0 := ℵ 1 .
Below we list some additional properties of these cardinals.Unless noted otherwise, proofs can be found in [Bla10].
2 Only the trivial inequality p ≤ t is used in this text.This work contributes to the project of constructing a forcing model satisfying: (♡) All the cardinals in Figure 2 are pairwise different, with the obvious (ZFC provable) exception of the dependent entries add(M) = min{b, cov(M)} and cof(M) = max{non(M), d}, and the Martin axiom numbers m, m k for some 2 ≤ k < ω, and m ω , which can not have more than one value >ℵ 1 , see Fact 1.2(7).In this direction [GKS19] constructed a forcing model, using four strongly compact cardinals, where all the ten (non-dependent) values of Cichoń's diagram are pairwise different (a situation we call Cichoń's Maximum), as in Figure 3(A).This was improved later in [BCM21] by only using three strongly compact cardinals; finally in [GKMS22] it was shown that no large cardinals are needed for Cichoń's Maximum.
A model of Cichoń's Maximum with the order as in Figure 3(B) was obtained in [KST19].Although this model initially required four strongly compact cardinals as well, the methods of [GKMS22] allow to remove the large cardinal assumptions also here.
As a next step towards (♡), [GKMS21] proved: ).Under GCH, for any k ∈ [1, ω), there is a cofinality preserving poset P k forcing that (a) Cichoń's Maximum holds with the order of Figure 3  corresponds to the model in [GKS19], and (B) to the model in [KST19] (both proven consistent in [GKMS22] without large cardinals).Each arrow can be < or = as desired.
In this paper, we continue this line of work by including, in addition, s and r.In both theorems above, item (b) can also be replaced by ℵ 1 < m ω < p < h < add(N ) while m k = ℵ 1 for all k < ω.Those are the only possible constellations of the Knaster numbers, by Fact 1.2(7), unless you count m as the 1-Knaster-number: In contrast to Theorem 1.3 (where we do not control r, s), we cannot force m > ℵ 1 with the methods we use here.We cannot just iterate over all small ccc forcings one by one to increase m, as our method requires that all iterands of the forcing iteration have to be "homogeneous".So instead of using a certain small forcing Q as iterand, we will use a finite support product over all variants as iterand.So only if Q (and therefore all variants) is Knaster, 3 this product can be used in a ccc iteration; accordingly we can increase the Knaster numbers but not m itself.

Main
We remark that the full power of GCH is not required in the Main Theorem, but we do need some assumption on cardinal arithmetic in the ground model.See details in Section 7.
In order to include s and r in our main result, we need a new preservation theorem for splitting families.Previously, the following was known in the context of FS (finite support) iterations: [BD85] Hechler forcing (for adding a dominating real) preserves splitting families witnessing the property LCU Rsp (κ) for any uncountable regular κ (see Section 3).
3 Or at least stays ccc in ccc extensions.
[JS88] Assuming CH, any FS iteration of Suslin ccc posets forces that the ground model reals form a splitting family.
In this paper we will use a splitting family obtained by a FS product of Hechlertype posets (cf.[Hec72]) which we call G B ; the support of G B is a graph B of size ℵ 1 with certain homogeneity properties.We then show that this splitting family is preserved by certain FS iterations, which we will call "symmetric Suslin-λ-small".
(Every FS iteration of Suslin ccc posets with parameters in the ground model is such an iteration, but our application will not use such "full" Suslin ccc forcings.)Similar preservation techniques have appeared in different contexts.For instance, concerning preservation of mad (maximal almost disjoint) families, Kunen [Kun80] constructed, under CH, a mad family that can be preserved by Cohen posets; afterwards, Steprans [Ste93] showed that, after adding ω 1 -many Cohen reals, there is a mad family of size ℵ 1 that can be preserved in further Cohen extensions; Fischer and Brendle [BF11] constructed a Hechler-type poset H A with support (any uncountable set) A that adds a mad family indexed by A, which can be preserved not only in further Cohen extensions but after other concrete FS iterations, thus generalizing Steprans' result because H ω1 = C ω1 ; [FFMM18,Mej19a] showed that any such mad family added by H A can be preserved by some general type of FS iterations, but the most general result so far was shown in [BCM21]: Any κ-Fr-Knaster poset preserves κ-strong-Md-families (with κ uncountable regular; the mad family added by H κ is of such type).
There are deep technical differences between the mad family added by this H A , and the construction of a splitting family in this paper: No structure is needed on A, and because of this it is clear that Hechler's posets satisfy H A ⋖ H B whenever A ⊆ B; but we cannot guarantee G B0 ⋖ G B for our posets, whenever B 0 is a subgraph of B. Also, G B itself does not add a splitting family, but it just adds a set of Cohen reals {η a : a ∈ B} over the ground model (recall that we do not have intermediate extensions by restricting the support B).Hence, the FS product (or iteration, which is the same, as the poset G B is absolute) of size κ of such posets adds a splitting family of size κ (witnessing LCU R (κ)) formed by the previously mentioned Cohen reals.It is clear that just adding κ many Cohen reals produces a splitting family satisfying LCU R (κ), but we need to use FS support products of κ many G B (with B of size ℵ 1 , instead of just one G B ′ with B ′ of size κ), and we need the graph structure on B, to be able to guarantee the preservation of the new splitting family.The forcing structure is very important here because an isomorphism of names argument is required for this preservation.
The strategy to prove the main theorem is similar to Theorem 1.3.We first show how to construct a ccc poset that forces distinct values for the cardinals on the left side of Cichoń's diagram, including some of the other cardinal characteristics (like s in this case).Afterwards, methods from [GKMS22,GKMS21] are applied to this initial forcing to get the poset for the main theorem.§3 The LCU and COB properties are reviewed from [GKS19, GKMS22, GKMS21].These describe strong witnesses to cardinal characteristics associated with a definable relation on the reals.Examples of such cardinal characteristics are the Cichoń-characteristics as well as s and r.
§4 We introduce the forcing G B , which has as support a suitable 2-graph B. We look at FS iterations of ccc posets, in general, whose initial part is a FS product of posets of the form G B where B is in the ground model.We define λ-small history iterations (where on a dense set, conditions have <λ-sized history), as well as symmetric iterations, and show that symmetric λ-small history iterations allow us to control s (and later also r).
§5 We define Suslin λ-small iterations, which are λ-small history iterations, and give consequences of this notions, as well as sufficient conditions to get symmetric ones.§6 Closely following [GKS19], we construct a symmetric Suslin-λ-small iteration P 0 that separates the cardinals on the left hand side of the diagram, with cov(M) = c and s = p.§7 We show how the tools of [GKMS22,GKMS21] can applied to P 0 , resulting in a forcing that gives the main theorem.

§8
We discuss some open questions related to this work.

Suitable 2-graphs
In this section we define and construct suitable 2-graphs.
Definition 2.1.Say that B := ⟨B, R 0 , R 1 ⟩ is a 2-edge-labeled graph, abbreviated 2-graph or bi-graph, if (i) R 0 and R 1 are irreflexive symmetric relations on B, (ii) R 0 ∩ R 1 = ∅.In other words: Between two nodes x and y there is at most one edge, with color 0 or 1.
Concerning 2-graphs, we define the following notions.
The 2-graph of Figure 4 does not have a coloring (with full domain) respecting it.
For an infinite cardinal λ, we say that B is a suitable bi-graph of size λ if (iii) for each e ∈ {0, 1}, there is a W e ⊆ B such that W e is a complete R e -graph (as a subgraph of ⟨B, R e ⟩), (iv) In the rest of this section we will show that suitable graphs of any infinite size exist.
Remark 2.3.In our applications, we only need the following weakening of property (v): for any t ∈ [B] <ℵ0 , a ∈ t and e ∈ 2, there is some η : t → 2 that respects B such that η(a) = e.The only place where (the weakening of) (v) is used is in the proof of Lemma 4.2(b).However, this weakening is equivalent to (v) itself by compactness.
Lemma 2.5.The relation ⪯ − is a partial order in the class of bi-graphs.Even more, if ⟨B α = ⟨B α , R α 0 , R α 1 ⟩ : α < γ⟩ is a ⪯ − -increasing sequence, then Proof.We show that ⪯ − is transitive (the rest is straightforward).So assume that B 0 ⪯ − B 1 and B 1 ⪯ − B 2 and, towards a contradiction, that there is a path This path may contain cycles of the form ⟨b j : j ∈ [j 0 , j 1 ]⟩ for some j 0 < j 1 such that . By removing such loops (and leaving the points in B 1 ), we may assume that the path ⟨b j : j ≤ n⟩ does not contain such loops.Since B 0 ⪯ − B 1 , we must have that the path ⟨b j : j ≤ n⟩ intersects B 2 ∖ B 1 , so we can find Theorem 2.6.For any infinite cardinal λ, there is a suitable bi-graph of size λ.
Proof.Our construction of the graph relies on the partial order P whose elements are tuples The order is defined by q ≤ p if , and (O4) for a ∈ B p and e ∈ {0, 1}, η q a,e extends η p a,e .By recursion, we construct a decreasing sequence ⟨p α : α ≤ λ⟩ in P , Our desired graph will be B λ .For the construction, we use a book-keeping bijection h : λ → 2 × 6 λ such that, whenever h(α) = (i, t), t k ≤ α for all k < 6.At each step α < λ, we enumerate B α = {b α,η : η < λ}.
Case i = 0: Let a 0 := b t0,t1 , a 1 := b t2,t3 and c := b t4,t5 , which are in B α .We aim to construct a p α+1 ≤ p α such that (a 0 , a 1 ) ∈ C α+1 and c ∈ D α+1 a0,a1 .Define p := ⟨B, R 0 , R 1 , C, f , η⟩ ∈ P as p α but with C := C α ∪ {(a 0 , a 1 )} and, in the case (a 0 , a 1 ) / ∈ C α , let f p a0,a1 be the only function from {a 0 } onto {a 1 } (notice that singletons are ⪯ − -below a bi-graph).Clearly p ∈ P and p ≤ p α .In the case c ∈ D p a0,a1 , let p α+1 := p.So assume that c / ∈ D := D p a0,a1 .We say that ⟨x ℓ : ℓ ≤ n⟩ is a path from c to D if it is a path in B such that x 0 = c, x n ∈ D and x ℓ / ∈ D when 0 < ℓ < n.The point x n is called the end-point of the path.Since B| D ⪯ − B, any two paths from c to D must go to the same endpoint, which we denote by z * if it exists (i.e. if a path from c to D exists).Let if ȳ = ⟨y j : j ≤ n⟩ is a path in B from y 0 ∈ D * to y n ∈ D * and y 0 ̸ = y n then: in the case y 0 , y n ∈ D, y j ∈ D for some 0 < j < n because B| D ⪯ − B; otherwise, the path can be extended (on both sides) to a path from c to D, so it lies inside D * .
Case i = 1: Let a 0 := b t0,t1 , a 1 := b t2,t3 and d := b t4,t5 , which are in B α .Proceed similarly as in the previous proof to construct p α+1 ≤ p α such that (a 0 , a 1 ) ∈ C α+1 and d ∈ E α+1 a0,a1 .In the limit step γ ≤ λ, we let p γ be the infimum of {p α : α < γ} in P .We show that this infimum exists (which shows that any decreasing sequence in P of length <λ + has an infimum, i.e.P is <λ + -closed).Let B γ be the ⪯ − -supremum of and η γ a,e := α<γ η α α,e (abusing notation in the last two unions, since formally the union starts at some α 0 from where the functions start to appear), which defines this infimum p γ .
At the end, our book-keeping functions and the construction ensures that Remark 2.7.In the previous proof, we could have argued a bit differently, using that the partial order P is <λ + -closed.In the successor step of the proof, we have actually proved that the following sets are dense in P for a, b, c ∈ λ + : }. Since P is <λ + -closed, we can find a filter G ⊆ P with p 0 ∈ G intersecting all the dense sets above.By taking unions of the components of the members of the filter (similar to the limit step of the proof above), we obtain a bi-graph B + = ⟨λ + , R 0 , R 1 ⟩ satisfying the conditions of a suitable bi-graph of size λ + , with the exception that W 0 and W 1 have size λ.Now, pick a large enough regular cardinal χ and an elementary submodel N ⪯ H χ of size λ with λ ∪ {B + } ⊆ N .Then, B := B + | λ + ∩N is a suitable bi-graph of size λ.

Cardinal Characteristics, COB and LCU
Many classical characteristics can be defined by the framework of relational systems as in e.g.[Voj93,Bla10].Say that R := ⟨X, Y, R⟩ is a relational system if X and Y are non-empty sets, and R is a relation.The following cardinal characteristics are associated with R. In this work, we are particularly interested in relational systems R such that (RS1) X and Y are subsets of Polish spaces Z 0 and Z 1 , respectively, and absolute for transitive models of ZFC (e.g. they are analytic); (RS2) R ⊆ Z 0 ×Z 1 is absolute for transitive models of ZFC (e.g.analytic in Z 0 ×Z 1 ).When these properties hold we say that R is a relational system of the reals.In all the cases explicitly mentioned throughout this paper, X and Y are Polish spaces themselves and R is Borel in X × Y .In this case, there is no problem to identify X = Y = ω ω , and we call R, or rather the characteristics b(R) and d(R), Blassuniform (cf.[GKMS21, §2]).Also all Cichoń-characteristics are Blass-uniform.The Blass-uniform relational systems we use for these characteristics are (as in the Cichoń's Maximum constructions) in some instances slightly different from the "canonical" ones.See e.g.[BCM21, Ex. 2.16], [Mej19b, Ex. 2.10] and [GKS19, §1] for the definition of the Blass-uniform relational systems corresponding to the Cichoń-characteristics.
As in [GKMS22] we also look at relational systems S = ⟨S, S, ≤⟩ where ≤ is an upwards directed partial order on S.Here cp(S) := b(S) is the completeness of S, and cof(S) := d(S) is the cofinality of S. Recall that, whenever S has no greatest element, cp(S) ≤ cof(S), and equality holds when the order is linear.
The following is a very useful notion to calculate the value of cardinal characteristics (specially in forcing extensions).Definition 3.2 (cf.[GKS19, §1]).Fix a directed partial order S = ⟨S, ≤⟩ and a relational system R = ⟨X, Y, R⟩.Define the property:

Cone of bounds. COB R (S) means: There is a family ȳ = {y
When L = ⟨L, ≤⟩ is a linear order, we additionally define Linear cofinally unbounded.LCU R (L) means: There is a family In the following remarks we address very natural characterizations and consequences of these properties.
Remark 3.3 (Tukey connections and COB).Let ȳ be a witness of COB R (S).By the definition of COB R (S) we have that the functions f : X → S and g : S → Y , defined by f (x) := i x and g(i) := y i , form a Tukey connection from R into S.So we conclude that COB R (S) holds iff R ≤ T S, where ≤ T denotes the Tukey order.
Remark 3.4 (Duality and LCU).Let R = ⟨X, Y, R⟩ be a relational system.The Although LCU is a particular case of COB, they are used with different roles in our applications, so it is more practical to use different notations.
As a direct consequence of these remarks: Let R be a relational system, S a directed partial order and let L be a linear order without greatest element.Then (a In our applications we aim to force COB R (S) and LCU R (L) for a given relational system of the reals R; this will help us compute the value of b(R) and d(R) in generic extensions.For this purpose, the following variation of Definition 3.2 is very practical.

Definition 3.6 ([GKMS22]
).Let R = ⟨X, Y, R⟩ be a relational system of the reals, S = ⟨S, ≤ S ⟩ a directed partial order, L = ⟨L, ≤ L ⟩ a linear order, and let P be a forcing notion.Define the following properties.

COB R (P, S):
There is a family ẏ = { ẏi : i ∈ S} of P-names of members of Y V P such that, for any P-name ẋ of a member of X V P there is some i ∈ S such that ⊩ P ∀j ≥ S i ( ẋR ẏj ).

LCU R (P, L):
There is a family ẋ = { ẋi : i ∈ L} of P-names of members of X V P such that, for any P-name ẏ of a member of Y V P there is some i ∈ L such that Remark 3.7.Concerning the properties COB R (P, S) and LCU R (P, L), the relational system R (i.e., both base sets as well as the relation) are interpreted in the generic extension (this is why we required these objects to be definable), while S and L are taken as sets in the ground model (not interpreted).
It is clear that COB R (P, S) implies ⊩ P COB R (S).Although the converse is not true in general, it holds in the cases we are interested in, when P is ccc and cp(S) is uncountable.More precisely, if cp(S) is uncountable and P is cp(S)-cc then COB R (P, S) is equivalent to ⊩ P COB R (S).Moreover, P forces cp(S) V P = cp(S) V and cof(S) V P = cof(S) V , so, by Lemma 3.5, in the generic extension COB R (S) implies cp(S) Likewise, LCU R (P, L) implies ⊩ P LCU R (L), and the converse holds whenever L has no greatest element, cof(L) is uncountable and P is cof(L)-cc.
However, the restriction "cp(S) is uncountable and P is cp(S)-cc" is not required for the following result.
Lemma 3.8 ([GKMS22, Lemma 1.3]).Let R be a relational system of the reals, S a directed partial order without greatest element, and let P be a forcing notion.

Preserving splitting families with symmetric iterations
4.A.The single forcing G B .Using suitable 2-graphs, we define a poset which will be used as factor for the forcing adding the splitting families we aim to preserve.
and aR e b, then {q(a, k), q(b, k)} ̸ = {e}.For a ∈ B denote by ηa the name of the generic real added at a, that is, G B forces that, for any k < ω, ηa (k) = e iff p(a, k) = e for some p in the generic set.
For • e ∈ {0, 1}, Proof.To see (a), first note that since |B ×ω| = ℵ 1 , by Engelking-Kar lowicz [EK65] there is a countable set H ⊆ 2 B×ω such that any finite partial function from B × ω into 2 can be extended by some member of H.
or pr a (p) is the empty sequence otherwise.It is enough to show that pr a is a forcing projection, that is, (i) for any p, q ∈ G B if q ≤ p then pr a (q) ⊇ pr a (p), (ii) for any p ∈ G B and s ∈ C, if s ⊇ pr a (p) then there is some q ≤ p in G B such that pr a (q) ⊇ s (even pr a (q) = s), (iii) pr a [C B ] is dense in C (even pr a is onto).Property (i) is easy, (ii) follows by Definition 2.1(v), and (iii) follows by (ii) and the fact that pr a (∅) = ⟨ ⟩.
We will use FS iterations where the first step is given by a FS product of posets of the form G B as above.It is clear that, if B is a S2G in the ground model, then it is still a S2G in any extension preserving ω 1 .On the other hand, constructing G B from B is absolute for transitive models of ZFC, so any finite support product of posets of the form G B is forcing equivalent to their finite support iteration (as long as the sequence of 2-graphs lives in the ground model).

4.B.
Suitable iterations, nice names and automorphisms.We now introduce some notions associated with these iterations, relevant for the preservation of splitting families.
From this point on, products of ordinals (such as ω 1 π) should be interpreted as ordinal products.
Definition 4.4.A suitable iteration is defined by the following objects: where the first iterand is the FS product of the G B δ for δ < π 0 , called P π 1 , and the following iterands are indexed by ξ ∈ π ∖ π 1 and are ccc posets called Qξ .
As usual, we denote with P ξ the result of the iteration up to ξ (for π 1 ≤ ξ ≤ π), and use P to denote either P π or the whole iteration (or its definition).See Figure 5 for an illustration.
Remark 4.5.Note that we could also view P π1 as (the result of) a FS-iteration of length π 0 (instead of length 1, as we do in the definition).Then we would get an iteration P of π 0 + (π − π 1 ).However, P π1 is not a FS iteration of length π 1 , at least not with natural iterands, see Remark 4.3.
Let us mention some notation: Notation 4.6.(1) A real-number-poset is a poset whose universe is a subset of the set of real numbers.For simplicity, we identify the "set of real numbers" with the power set of ω.
. A suitable iteration.π 1 = ω 1 π 0 is partitioned into π 0many intervals of length ω 1 , and B δ := [ω 1 δ, ω(δ + 1)), the set of vertices of the graph B δ , is the δ-th interval of this partition.A suitable iteration is a FS product of the G B δ for δ < π 0 , followed by a FS iteration of ccc posets.The iterands of the FS iteration that follow are indexed by α ∈ [π 1 , π).
(2) For notational simplicity we will often identify P ζ+1 (a set of partial functions) with P ζ * Qζ (a set of pairs (p, q) with p ∈ P ζ and p ⊩ q ∈ Qζ ).
(3) Similarly, we will not distinguish between sequence of names and names of sequences.
We now define the "support" supp(p) ⊆ π of a condition p (as opposed to the domain dom(p), which is, as we are dealing with a FS iteration, a finite subset of the index set {0} ∪ (π ∖ π 1 )).We will also define the "history" H of a name and of a condition: Definition 4.7.Let P be a suitable iteration.
reader prefers a different formal definition of FS iteration, then they should modify the definition of H to make sure that H(x) = ∅.
Remark.H is not a "robust" notion: ⊩ τ = τ ′ does not imply H(τ ) = H(τ ′ ).Still, it is a very natural and useful notion, which has appeared (in slightly different contexts) many times in forcing theory: If τ is a P π -name, then H(τ ) ⊆ π is the set of coordinates the name τ "depends on", more concretely, τ can be calculated (by a function defined in V ) from the sequence of generic objects at the indices in H(τ ).In the case of FS iterations where all iterands are real-number-posets (as in [She00,GKS19]), H(p) is countable for p in a dense set; and "hereditarily nice names" for reals will also have countable history.In this paper we have to use hereditarily <λ-names (even for nice names of reals), the reason is indicated in Remark 4.16.
Let us fix some notation regarding the well-known "nice names": Definition 4.8.Let A and B be subsets of P.
(1) A P-name ṙ is a nice name for a subset of ω, determined by A, if ṙ has the form n∈ω {(ň, q) : q ∈ A n }, where each A n is a (possibly empty) antichain in P, and A = n∈ω A n .
(2) Analogously, Q is a nice name for a real-number-poset of size <λ , determined by B, if there is a µ < λ such that Q is a sequence ⟨ ṙi ⟩ i∈µ of nice names for subsets of ω determined by A i , together with a sequence ⟨ ẋi,j ⟩ i,j∈µ of nice names for elements in {0, 1} depending on an antichain A ′ i,j (where ẋi,j = 1 codes It is well known that every name of a subset of ω has an equivalent nice name.Moreover, as we can choose the conditions of the antichains in any given dense set, we get the following: Every automorphism of B induces an automorphism of G B , see Lemma 4.2(e).Therefore, a π 0 -sequence h of such automorphisms induces an automorphism of the (FS) product P π1 .Such an automorphism can sometimes be naturally extended to the whole iteration P (which will allow isomorphism-of-names arguments and subsequently show LCU sp ).
What do we mean by "naturally extend"?Recall that, whenever f : P → P is an automorphism on some poset P , and τ is a P -name, f sends τ to the P -name 7 A nice name ẋ of a member of {0, 1} depending on an antichain C ⊆ P (allowed to be empty) has the form ẋ = {( 0, p) : p ∈ C}.Note that p ⊩ ẋ = 1 for all p ∈ C, and q ⊩ ẋ = 0 for any q ∈ P incompatible with all the members of C.Moreover, H( ẋ) = p∈C H(p).
Also, (f −1 ) * (f * (τ )) = τ ; and p ⊩ φ(τ ) iff f (p) ⊩ φ(f * (τ )) whenever p ∈ P and φ(x) is a formula.If Q is a P -name and P ⊩ f * ( Q) = Q, then we can certainly extend f to P * Q.We say that P is h-symmetric, if this is the case in all steps of the iteration: Definition 4.11.Let P be a suitable iteration.
(1) A bijection h : In isomorphism-of-names arguments it is relevant to know when a condition or a name remains unchanged after applying an automorphism ĥ.The following states a sufficient condition: Lemma 4.12.Assume that P is h-symmetric and π 1 ≤ ξ ≤ π.
Proof.We show the three statements by induction on ξ.
For (a), we use a case distinction: Assume ξ = π 1 .If p ∈ P π1 then H(p) = supp p, and whenever h is the identity on supp p, it is clear that ĥπ1 (p) = p.The limit step is also immediate (there are no new conditions, and for names use ∈-induction).
For the successor step ξ = ζ + 1, assume p ∈ P ζ+1 and that h is the identity on For (c), the steps ξ = π 1 and ξ > π 1 limit are easy, so we deal with the successor step ξ = ζ + 1.So assume that ĝζ is defined and ĝζ = ĥ−1 ζ .Since ĥξ is defined,

4.C.
A digression: Self-indexed products.How to construct a symmetric iteration P? We have to make sure that at each step ζ the iterand Qζ is invariant under ĥ for all h ∈ H * .One case that will be useful: Qζ is a (ccc) FS product such that whenever Q is one of the factors, then ĥ * ( Q) is also one.But there is a technical difficulty here: We need ⊩ P ζ ĥ * ( Qζ ) = Qζ (i.e., really equality, not just isomorphism; as we want to get an actual automorphism of P ζ+1 ).This is not possible if we "naively" index the product with an ordinal.For example, assume Q0 , Q1 are such that ⊩ P ζ ĥ * ( Qi ) = Q1−i ̸ = Qi .Then Q0 × Q1 (the product So instead, we define (in the extension) the FS product F of a set F of posets as the set of all finite partial functions p from F into F satisfying p(Q) ∈ Q for all Q ∈ dom(p).We call this object the self-indexed product of the set F.
In our framework, we start with a ground model set Ξ ζ of P ζ -names of posets.In the P ζ -extension we let F be the set of evaluations of the names in Ξ ζ , and let Qζ be the self-indexed product of F.
Assume that all automorphisms from H * can be extended up to ζ. 8 We assume that Ξ ζ is closed under each h ∈ H * , i.e., Q ∈ Ξ ζ implies ĥ * We additionally assume that each factor is (forced to be) a real-number-poset.Assume that (p, q) ∈ P ζ * Qζ .We can densely assume that p decides the finite domain of q, more specifically, the 9 finite set y ⊆ Ξ ζ such that p forces that dom(q) is (the set of evaluations of) y.Also, for each Q ∈ y, we can assume that q( Q) is a nice name for a real, determined by some A Q. As usual, we can use a given dense set D ⊆ P ζ instead of P ζ .
For later reference: Fact 4.14.Assume that Qζ is a "self-indexed" product of Ξ ζ as described above, that each factor is forced to be a set of reals, and that D ⊆ P ζ is dense.If (p, q) ∈ P ζ * Qζ , then there is a (p ′ , q ′ ) ≤ (p, q) such that p ′ ∈ D decides the (finite) dom(q ′ ), 8 For this part all the properties of H * are not required; it is just enough that h ∈ H * implies h −1 ∈ H * . 9Or rather: a finite set, as different names in Ξ ζ might evaluate to the same object, i.e., index.
and each q ′ ( Q) is a nice name determined by some A Q ⊆ D. So in particular Remark 4.16.This is the reason hereditarily countable nice names are not sufficient in our setting to describe reals: Even the index Q in such a product Qζ is too complicated.However, as all the self-indexed products Qζ we use will have factors Q of size <λ, it turns out we can restrict ourselves to hereditarily <λ-names (this will be the dense set P * ζ of Definition 5.3).4.D. Symmetric small history iterations preserve splitting families.We are finally ready to prove the central fact about preservation of splitting families.Definition 4.17.Let λ be an uncountable cardinal.
(2) A suitable iteration P has λ-small history if, for any p ∈ P, there is a λ-small q ≤ p.
So in particular if P has λ-small history and ẋ is a name of a subset of ω, then there is an equivalent nice name ḃ which only uses λ-small conditions; and if µ ≥ λ has uncountable cofinality, then Theorem 4.19.Let P be a symmetric suitable iteration with λ-small history.Assume ℵ 1 ≤ λ ≤ µ ≤ π 0 are cardinals with µ regular.Then LCU Rsp (P π , µ) holds, and it is witnessed by { ηω1δ : δ < µ}.
Proof.Towards a contradiction, assume that there are p ∈ P and a P-name ḃ of an infinite subset of ω such that p ⊩ |{δ < µ : ηω1δ ↾ ḃ is eventually constant}| = µ.

Suslin-λ-small iterations
We now investigate suitable iterations where the iterand Qζ at step ζ > 0 (i.e., after the initial FS product) is (1) either a restricted (also called: partial) Suslin ccc poset (e.g., random forcing evaluated in some V P − ζ for some complete subforcing P − ζ of P ζ ); (2) or the FS product of (in our application: at most |π|-many) <λ-size posets of reals.More formally: Definition 5.1.Let λ be an uncountable cardinal.A Suslin-λ-small iteration (abbreviated Sλs) is a suitable iteration P with the following properties: (S1) π ∖ π 1 is partitioned into two sets Σ and Π. (S2) For ξ ∈ Σ, (i) P − ξ is a complete subposet of P ξ , (ii) S ξ is a definition of a Suslin ccc poset (with parameters in the ground model), (iii) Qξ is a P ξ -name for (S ξ ) V P − ξ .(S3) For ξ ∈ Π, (i) Ξ ξ is a set in the ground model, (ii) each element of Ξ ξ is a P ξ -name Q for a poset of size10 <λ consisting of reals, (iii) Qξ is (the P ξ -name for) the FS product of Ξ ξ .
Remark 5.2.Regarding (S3), recall that our setting requires Qξ (to be forced by P ξ ) to be ccc (as suitable iterations have to be ccc).In contrast, in (S2), Qξ will be always ccc "for free" (in V P ξ as well as in V P − ξ ), as it is an evaluation of a Suslin ccc definition (see [JS88]).
We now show that we can replace such an iteration ⟨P ′ ζ , Q′ ζ : ζ ∈ π⟩ with an isomorphic version ⟨P ζ , Qζ : ζ ∈ π⟩: The only difference will be in steps ζ ∈ Π, where we select (hereditarily) nice names for the factors Q ∈ Ξ ζ and make sure that Qζ is self-indexed.In addition, we will define a dense subset P * of hereditarily λsmall conditions, an extended "refined history domain" π + , and a "refined history" H * : P * → P(π + ).These are formalized in the following notions.
In addition, if p ∈ P * ξ+1 then the following is satisfied: (iii) When ξ > π 1 is limit and p ∈ P * ξ , then H * (p) has already been defined (because p ∈ P ζ for some ζ < ξ).(iv) For any P * ξ -name τ define, by ∈-recursion, Tidy Sλs iterations have many features that ease its manipulation, in particular, they have λ-small history.ζ -name for a real-number-poset of size < λ, it is determined by some B Q of size < λ.Hence so we can deduce (b).The limit step is immediate.□ As promised, we show that any Suslin-λ-small iteration is isomorphic to a tidy one.
Proof.By recursion on π 1 ≤ ξ ≤ π we construct the tidy iteration up to P ξ , along with its components, and the isomorphism i ξ : P ′ ξ → P ξ .We also guarantee that i ξ extends i ζ for any π 1 ≤ ζ < ξ.
which is clearly a complete subforcing of P ζ , and evaluate S ζ accordingly.Note that i ζ can be extended to an isomorphism i ζ+1 : P ′ ζ+1 → P ζ+1 in a natural way.
We define P * ζ+1 as the set of pairs (p, q) where p ∈ P * ζ , q is a P * ζ nice name for a real, and p ⊩ q ∈ S ζ ∩ V P − ζ .This is dense according to Fact 4.10(a).
Q′ ) is forced by P ζ to have size <λ, according to Fact 4.10(b), there is an equivalent ζ of size <λ.Let Ξ ζ be the set of all these names, and define Qζ to be the self-indexed FS product of the Q in Ξ ζ .We can obtain the isomorphism i ξ+1 in a natural way.
In our applications, P − ξ has the following form.Definition 5.8.Let P be a tidy Sλs iteration.For any X ⊆ π + , define P * ↾X := {p ∈ P * π : H * (p) ⊆ X}.Note that generally P * ↾X will not be a complete subforcing of P * ; but we will only be interested in the case where it is, see Lemma 5.15.
Of course we will have to also make sure that the assumption "P is a tidy Sλsiteration" is satisfied.The nontrivial points of these assumptions are: (In our case this will be trivial, as all factors Q will be Knaster).(II-b) For ζ ∈ Σ, P * ↾C ζ is a complete subforcing of P * .
We will see in Lemma 5.15 how to achieve this.
Proof.By induction on ξ ∈ [π 1 , π] we show that ĥξ can be defined for any h ∈ H * (towards (a)), and that (b) is valid for any closed C ⊆ ξ + .12Case ξ = π 1 : It is clear that ĥπ1 can be defined; ζ (as S ζ only uses parameters from the ground model), which is Qζ by induction hypothesis (as we assume that C ζ is closed).So we can extend ĥζ to ĥζ+1 .
We conclude that ĥξ [P * ↾C] ⊆ P * ↾C, but equality holds because the same is true for h −1 .
Case ξ limit: By induction hypothesis, ĥζ is defined for all ζ < ξ, so ĥξ is defined (as its union).On the other hand, if Proof.Property (a) is straightforward.We show by induction on ξ H * ( Q) has size <λ ≤ µ by Lemma 5.5(c), and therefore also the set A * has size <µ.

It is clear that
Since H * is a group, {ζ} ∪ X is closed.We claim that we get the same set X if we replace H * with H ′ := {g ∈ H * : supp(g) ⊆ A * }.As H ′ has size <µ (recall that | supp(g)| ≤ ℵ 1 for any g ∈ H * ), we get |A| < µ, as required.
(b) Let ⟨B i : i ∈ λ⟩ be an increasing sequence of closed subsets of π + such that P * ↾B i is a complete subforcing of P * .Set B := i∈λ B i .According to 5.13(a) B is closed.Assume that A ⊆ P * ↾B is a maximal antichain.Any p, q ∈ A are incompatible in P * ↾B i for some i, and therefore in P * .Due to ccc, A is countable, and by Corollary 5.10(b) there is an i < λ such that A ⊆ P * ↾B i .Therefore A is maximal in P * .□ Corollary 5.16.With the same hypothesis of Lemma 5.15, if P ⊆ P * π has size <µ, then there is some P − ⋖ P * π of size <µ such that P ⊆ P − .Proof.Apply Lemma 5.15 to A := p∈P H * (p). □ We now summarize what we already know about the construction that we are going to perform in the next section: Corollary 5.17.Let λ be regular uncountable and assume λ ≤ |π|, |π| <λ = |π|, and that π ∖ π 1 is partitioned into Σ and Π.We inductively construct a (tidy) Sλs iteration P as follows: ( Proof.Each Qζ is forced to be ccc (by either absoluteness or the Knaster assumption), so we get a valid iteration (and we assume that we choose the names for the iterands such that we get a tidy Sλs-iteration).
LCU follows from Lemmas 5.14 and 5.12(a), and Theorem 4. In this section, we prove the first step of the main theorem: Theorem 6.11, which gives the independence results for the left hand side.After the work we have done in the previous sections, this is basically a simple variant of the construction in [GKS19].
As in [GKS19, GKMS22], for i ∈ {1, 2, 3, 4, sp} consider Blass-uniform relational systems For completeness, we review the posets we use in our construction.Definition 6.2.Define the following forcing notions (where the forcing in item (i) is designed to increase b i ): (1) Amoeba forcing A is the poset whose conditions are subtrees T ⊆ 2 <ω without maximal nodes such that [T ], the set of branches of T , has measure < 1 2 (with respect to the Lebesgue measure of 2 ω ).The order is ⊇.
For each of the posets above it is easy to construct a 1-1 function from the poset into ω ω .So, until the end of this section, the posets above are seen as subsets of ω ω .Moreover, the posets (1)-(4) are Suslin ccc, and they are homeomorphic to a Borel subset of ω ω (and the order is Borel as well).
In the proof of Theorem 6.6 we deal with special restrictions of the posets (1)-(4) under sets of reals of the following form.Definition 6.3.Let λ ≥ ℵ 1 be a cardinal.Say that E ⊆ ω ω is λ-elementary if E = ω ω ∩ N for some regular χ ≥ (2 ℵ0 ) + and some N ⪯ H χ of size <λ, where H χ denotes the collection of hereditarily <λ-size sets.
We look at posets of the form S ∩ E where S is a poset as in (1)-( 4) and E ⊆ ω ω is λ-elementary.Note that, whenever χ ≥ (2 ℵ0 ) + , N ⪯ H χ and E = ω ω ∩ N , we have S ∩ E = S N .Therefore: We now show how to modify the forcing construction in [GKMS21, §4 & §5] to include LCU sp and COB sp , by performing a construction according to the previous section, in particular to Corollary 5.17.We will assume the following: Our intention is to show the following: Goal 6.6.There is a ccc poset P of size λ 5 such that, for any i ∈ {1, 2, 3, 4, sp}, (a) LCU i (P, θ) holds for any regular λ i ≤ θ ≤ λ 5 .(b) There is some directed S i with cp(S i ) = λ i and |S i | = λ 5 such that COB i (P, S i ) holds.(c) P forces p = s = λ sp and c = λ 5 .
as in Definition 6.3.Note that S i , and therefore also every variant S i ∩ E, is linked and therefore Knaster.First let us note that sometimes it is more convenient to view P as a FS ccc iteration, where we first add the λ 5 -many G B forcings (of size ℵ 1 ), then the λ 5many Cohen reals, and then the rest of the iteration, where we interpret each FS product Q ζ for ζ ∈ Π i as a FS iteration with index set λ 5 = |Ξ i |.So all in all we can represent P as a FS iteration (6.7) For each α < λ 5 + λ 5 , | Q′ α | ≤ ℵ 1 , and for each α ≥ λ 5 + λ 5 in δ ′ , we say that Q′ α "is of type i" for i ∈ {m, p, 1, 2, 3, 4}, if either Q′ α = Qζ for the respective ζ ∈ Σ i , or if Q′ α is a factor Q of Qζ for the respective ζ ∈ Π i .Note that P ′ λ5 = P π1 .Lemma 6.8.The construction above satisfies Goal 6.6, apart from possibly LCU 3 .
Proof.c = λ 5 , as we use a construction following Corollary 5.17.Item (a) for i = sp, i.e., LCU sp : This also follows from Corollary 5.17, and implies ⊩ P s ≤ λ. p = s = λ: To see p ≥ λ it is enough to show (in fact, equivalent, by Bell's theorem [Bel81]): For every σ-centered poset Q ′ of size <λ (and contained in ω ω ), and any collection D of size <λ of dense subsets of Q ′ , there is a Q ′ -generic set over D. Any such Q ′ and D are forced to be already in the P α -extension for some α < π.Pick some ζ ∈ Π p larger than α.Then a name Q for Q ′ is used as factor of P ζ , i.e., in P ζ+1 there is a Q-generic object (over D).
ZFC shows p ≤ s, and as s ≤ λ we get equality.
• If Q ′ α is of type 2, then it is a subalgebra of the measure algebra, and thus (R LCU 1 , ℵ 1 )-good.For (R LCU 2 , λ 2 )-good the argument is even simpler: All Q ′ α have size <λ 2 or are σ-centered; and for (R LCU 4 , λ 4 )-good the argument is trivial, as all Q ′ α have size <λ 4 .
So this argument shows that, in the intermediate model V Pπ 1 , the rest P ′ of the forcing satisfies LCU i (P ′ , λ i ), witnessed by the Cohen reals {η α : α ∈ [π 1 , π 1 + λ i )}.This implies by definition of LCU that in the ground model LCU i (P, λ i ) holds, witnessed by the same Cohen reals.Item (b) for i ∈ I 1 , i.e., COB i : This is also basically the same as in [GKS19, §1.2],where this time we argue from the ground model V , not the intermediate model V Pπ 1 .We define the partial order S i to have domain Σ i , ordered by , and our book-keeping ensures that S i is <λ i -directed.Corollary 5.10(c) together with the fact that λ ≤ λ i shows that our bookkeeping will catch every real in the P-extension.Therefore S i , and the generics added at stages in S i , witness the COB property.Item (b) for i ∈ I 0 ∩ {1, 2, 3}: This is very similar: Let S i be the set of pairs (ζ, Ė) such that ζ ∈ Π i and Ė is a nice P * ζ -name of a λ i -elementary subset of ω ω .We order S i as follows: (ξ 1 , Ė1 ) ≤ i (ξ 2 , Ė2 ) iff ξ 1 ≤ ξ 2 and the empty condition forces that Ė1 ⊆ Ė2 .
And for any <λ i -sequence ⟨E j : j ∈ J⟩ of nice names for λ i -elementary sets E j we can find a nice name for a λ i -elementary set E ⊇ j∈J E j .This shows that S i is <λ i -directed.Item (b) for i = sp, i.e., COB sp : This is basically the same: Among the σ-centered forcings that we use as factors in step ζ of type p, there are Mathias-Prikry forcings M Ḟ on (free) filter bases of size <λ.In more detail: Assume Ḟ is a P * ζ -name for a filter base of size <λ, so set Q := M Ḟ .Then Q is σ-centered and adds a real which is not split by any set in Ḟ .

Item (d) is exactly the same as in [GKMS21, Lemma 4.7]. □
To guarantee b ≤ λ 3 , we have to make sure that the large iterands (i.e., the forcings of size ≥ λ 3 ) do not destroy LCU 3 (small forcings are, as usual, harmless).In our construction, the only large forcings are the partial eventually different forcings at steps ζ ∈ Σ 4 .For these forcings, we introduce in [GKS19] (based on [GMS16]) ultrafilter-limits and use them to preserve LCU 3 .The same argument works here.Remark 6.9.Note that in the proof of Lemma 6.8 we do not require the hypotheses χ = χ <χ and λ 3 = χ + from Assumption 6.5.These will be used to guarantee LCU 3 in the following subsection.
We work from the ground model, not in the intermediate P π1 -extension.Accordingly, we have to incorporate the initial segment of the iteration P π1 = P ′ λ5 into the argument.This is no problem, as we just have to deal with another type of small forcing, the Q′ α for α < λ 5 , which all have size ℵ 1 .Of course, E ′ := E V P * ζ ↾C ζ is closed under conjunctions of conditions, i.e., satisfies the assumptions of [GKS19, Fact 1.25].And instead of "ground model code sequences" we use "nice P * ζ ↾C ζ -names".The crucial part of the old proof is [GKS19, Lemma 1.30(d)].There, we use the notation w α ⊆ α, and Q α are those E-conditions that can be calculated in a Borel way from the generics with indices in w α , i.e., Q α = E ∩ V P * ↾wα ; and we show that the set of "suitable" w α is an ω 1 -club in [α] <λ4 , where "suitable" means: If we have a ground-model-sequence of (nice) Q α -names, then the D ε α -limit (a well-defined condition in eventually different forcing) is also element of Q α (for all ε ∈ χ).
The same argument gives us the following for our new framework: We can perform the construction of Lemma 6.8 and, at all indices ζ of type 4, the set of "suitable" The same argument as in the old proof (Lemma 1.31 there) then shows: Whenever all C ζ as chosen "suitably" (for all ζ of type 4), we get LCU 3 .□ Theorem 6.11.Assumption 6.5 is enough to find a P as required for Goal 6.6.
Proof.Let R be the poset of partial functions r : χ × λ 5 → {0, 1} with domain of size <χ (ordered by extension).As we assume χ <χ = χ, this poset is χ + -cc, and obviously <χ-closed, so it does not change any cofinalities.As in the old proof, at each step ζ of type 4 in the inductive construction of P, we can go into the Rextension of the ground model, use Lemma 6.10 to get a suitable C 0 ζ (above some initial set given by the usual book-keeping), find in V some C0 ζ such that C 0 ζ is forced to be a subset.Now we iterate this λ many times (not just ω 1 as in the old proof), taking unions at limits, and use the fact that the "suitable" parameters C ζ are closed under λ-unions (they form a λ-club in [ζ + ] <λ4 ).
This way we get a sequence of parameters C ζ in the ground model, such that if we define in the R-extension a forcing P ′ using these parameters we get LCU 3 (P ′ , κ); a simple absoluteness argument [GKS19, Lemma 1.33] then shows that these parameters will already define in V a forcing P with LCU 3 (P, κ).
Note: We do not interpret Ξ ζ (for ζ ∈ Π) in the R-extension, but use it with the same meaning it has in V .So P ′ may not be symmetric in the R-extension, but this is not important here: We are only interested in LCU 3 (P ′ , κ) in this argument, and we do not claim that P ′ in the R-extension satisfies the other properties we have already shown for P.And for LCU 3 (P ′ , κ), any iterand that has size <λ 3 is unproblematic.□ Remark 6.12.It is not necessary to restrict λ 3 to a successor cardinal in Assumption 6.5.To allow regular λ 3 in general, we forget about χ in Assumption 6.5 and just assume that λ <λ3 3 = λ 3 > ℵ 1 .In this way, Lemma 6.10 is valid by assuming 2 λ3 ≥ λ 5 instead, and Theorem 6.11 is true when replacing χ by λ 3 in the proof (i.e., R gets modified and it forces 2 λ3 ≥ λ 5 ).No further changes in the proofs (even in those from [GKS19]) are needed to justify this.
On the other hand, can we allow λ 3 = ℵ 1 in Assumption 6.5? (So all cardinals except λ 4 and λ 5 are ℵ 1 .)Although we can make the construction in this case, now the forcings G B δ have size λ 3 = ℵ 1 , so they could destroy LCU 3 (P, ℵ 1 ).An alternative to deal with this problem is to perform a similar iteration with π 0 = 0 (so π 1 = 0, that is, no initial FS product of G B is used) and guarantee LCU R * (P, κ) for any regular κ ∈ [ℵ 1 , λ 5 ] with the methods of this subsection (i.e. the methods from [GKS19, §1.3 & §1.4]) adapted to R * , where R * is the Blass-uniform relational system from [KW96] (see also [Mej13,Example 2.19]) such that b(R * ) = max{b, s} and d(R * ) = min{d, r}.
6.D.The other constellations for the Knaster numbers.So far we have assumed that λ m > ℵ 1 and that k 0 < ω.We now remark on how to prove the other cases: Case λ m = ℵ 1 .We only change I 0 := {p} ∪ {i ∈ [1, 4] : λ i ≤ λ} (so (I2)(i) is excluded in the construction).Check details in [GKMS21, Lemma 4.7].Note that here the value of k 0 is irrelevant.
Case k 0 = ω and λ m > ℵ 1 .Force with P cal,λm * P where P cal,λm is the precaliber ℵ 1 poset from [GKMS21, §5] and P is the forcing resulting from the construction above (in the P cal,λm -extension).17 6.E.The alternative order of the left side.The construction of [KST19,§2] for the alternative order of the left side of Cichoń's diagram can also be adapted in the situation of the previous theorems.This is just interchanging the order of the values of b and cov(N ), that is, instead of forcing cov(N ) = λ 2 ≤ b = λ 3 , we force b = λ 3 < cov(N ) = λ 2 .See also [Mej19b] for the weakening of the hypothesis GCH: Theorem 6.13.Theorem 6.11 (and Goal 6.6) is still valid when, in Assumption 6.5, we replace λ 1 ≤ λ 2 ≤ λ 3 < λ 4 by λ 1 ≤ λ 3 < λ 2 < λ 4 . 18Remark 6.12 also applies in this situation.

15 values
In this section, we review some tools from [GKMS22, GKMS21] and show how they are used to control the cardinal characteristics other than s.We describe the forcing constructions but we omit the details in the proofs, since these are exactly as in the cited references.
We use the notions of m-like cardinal characteristic and h-like characteristic from [GKMS21, §3].We do not need to recall their definition, but we only need some of their properties and to know that the cardinals m k (1 ≤ k ≤ ω) are m-like, h and g are h-like, and p and t are of both types.
Lemma 7.1 ([GKMS21, Cor.3.5]).Let κ be an uncountable regular cardinal, λ a cardinal, x a cardinal characteristic, and let P be a κ-cc poset that forces x = λ (so λ is a cardinal in the P extension).If M ⪯ H χ (with χ a large enough regular cardinal) is <κ closed and contains (as elements) P, κ, λ and the parameters of the definition of x, then P ∩ M is a complete subposet of P and: Lemma 7.2 ([GKMS21, Lemma 6.3]).Assume: (1) κ ≤ ν are uncountable regular cardinals, P is a κ-cc poset.
(3) For some relational systems of the reals R 1 i (i ∈ I 1 ) and some regular λ 1 i ≤ µ: P forces LCU R 1 i (λ 1 i ) (4) For some relational systems of the reals R 2 i (i ∈ I 2 ), and some directed order S 2 i with b(S 2 i ) = λ 2 i ≤ µ and |S 2 i | ≤ ϑ 2 i ≤ µ: P forces COB R 2 i (S 2 i ).
We are now ready to prove the main result of this paper.We use Notation 6.1 and the following assumption for all the results in this section. 18As in [KST19, §2], the relational system R LCU 2 corresponding to this result is not the same as the one for Theorem 6.11.Although this is a relational system of the reals, it is not Blass-uniform.ordered by end extension, and let P 1 be the poset constructed in Theorem 7.4.Exactly as in the proof of [GKMS21, Thm.7.4], P := P 1 × Q is as required.□ In the same way, we can prove the Main Theorem corresponding to Figure 3(B).In this case, we initial forcing P 0 is obtained from Theorem 6.13.Theorem 7.6.Both Theorems 7.4 and 7.5 are valid when Assumption 7.3 is modified in the following way: (i) We replace the order of the regular cardinals in (1) by (ii) In (3), we consider i 0 ∈ {0, 1}, but µ sp ∈ [µ 1 , µ 3 ] and µ r ∈ [µ 6 , µ 8 ] when i 0 = 1.(iii) In (4), instead of θ 3 = χ + 3 and χ <χ3 3 = χ 3 , assume θ 5 = χ + 5 and χ <χ5 5 = χ 5 .

Discussions
One obvious question is: Question 1.How to separate additional cardinals from Figure 2?
Another one: Question 2. How to get other orderings, where non(M) > cov(M)?
This is not possible with FS ccc iterations, as any such iteration whose length has uncountable cofinality δ forces non(M) ≤ cof(δ) ≤ cov(M), so alternative methods are required.A creature forcing method based on the notion of decisiveness [KS09, KS12] has been developed in [FGKS17] to separate five characteristics in Cichón's diagram, but this method is restricted to ω ω -bounding forcings, i.e., results in d = ω 1 .An unbounded decisive creature construction might be promising.Alternatively, Brendle proposed a method of shattered iterations 20 , which also may be a way to solve this problem.Question 3. Are our main results (specifically, Theorems 6.11, 6.13, 7.4, 7.5 and 7.6) valid for k 0 = 1?I.e., can we force m > ℵ 1 ?
For k 0 ≥ 2 there was no problem to include, in our iterations, FS products of k 0 -Knaster posets since they are still k 0 -Knaster (hence ccc), but we cannot just use FS products of ccc posets because they do not produce ccc posets in general.In particular, we do not know how to modify Theorem 6.11 to force m > ℵ 1 .
Question 4. Is it consistent with ZFC that b < s < non(M) < cov(M)?In this paper s ≤ b; and forcing s > b is much more difficult, since Mathias-Prikry posets may add dominating reals.Shelah [She84] proved the consistency of b = ℵ 1 < s = c = ℵ 2 by a countable support iteration of proper posets.Much later, Brendle and Fischer [BF11] constructed an FS iteration via a matrix iteration to force ℵ 1 < b = κ < s = c = λ for arbitrarily chosen regular κ < λ.However, in this latter model, non(M) = cov(M) = c.It is not clear how to adapt Brendle's and Fischer's methods to our methods and produce a poset for the previous question.

Figure 1 .
Figure 1.Cichoń's diagram (left).In the version on the right, the two "dependent" values add(M) = min{b, cov(M)} and cof(M) = max{non(M), d} are removed; the "independent" ones remain (nine entries excluding ℵ 1 , or ten including it).An arrow x → y means that ZFC proves x ≤ y.

Figure 2 .
Figure 2. Cichoń's diagram and the Blass diagram combined.An arrow x → y means that ZFC proves x ≤ y.

Figure 3 .
Figure 3.The two known consistent orders where all the (nondependent) values in Cichoń's diagram are pairwise different.(A) corresponds to the model in [GKS19], and (B) to the model in [KST19] (both proven consistent in [GKMS22] without large cardinals).Each arrow can be < or = as desired.
Theorem.Under GCH, for any k ∈ [2, ω) there is a cofinality preserving poset forcing that the cardinals in Cichoń's diagram, m k , p, h, s and r are pairwise different.More specifically: (a) Cichoń's Maximum holds, in either of the orders of Figure 3.(b) ℵ 1 = m k−1 < m k = m ω < p < h < add(N ).(c) s can assume any regular value between p and b.(d) r can assume any regular value in the dual position to s. E.g., if s < add(N ), then r can be any arbitrary regular in [cof(N ), c] (see details in Section 7).

Figure 4 .
Figure 4.A finite 2-graph which cannot be respected by any coloring.
Example 3.1.([Voj93, 2.2.2] or [Bla10, §4 & §5]) The splitting number s and the reaping number r are Blass-uniform: Denote R sp := ⟨2 ω , [ω] ℵ0 , R sp ⟩ where xR sp y iff x↾y is constant except in finitely many points of y.Then s = b(R sp ) and r = d(R sp ). 4 1}, a, b ∈ F p and aR e b, then ηa (k) and ηb (k) cannot both be e at the same time.(d) Assume for i ∈ {1, 2}: (b): Consider Cohen forcing C := 2 <ω ordered by end-extension.For a ∈ B define pr a : G B → C such that, for any p ∈ G B , pr a (p) := ⟨p(a, k)

Fact 4 .
10. (As P is ccc) Let D ⊆ P be dense and let λ be a cardinal with uncountable cofinality.(a) For any P-name of a real there is an equivalent nice name determined by A ⊆ D with |A| ≤ ℵ 0 .(b) For any name of a poset of size <λ consisting of reals, there is an equivalent nice name determined by a set B ⊆ D with |B| < λ.
19.For the size of the continuum, we use Corollary 5.10(d) to show by induction that |P * ξ | ≤ |π|: Assume this already is the case for ζ ∈ Π, then Ξ ζ consists of nice P * ζ -names for <λ-sized real-number-forcings, and there are only |P * ζ | <λ ≤ |π| many such nice names.□ 6.The forcing construction for the left hand side

Fact 6. 4 .
Let E ⊆ ω ω be elementary.Then: (1) The poset A ∩ E adds a (code of a) Borel measure zero set that contains all Borel null sets with Borel code in E. (2) The generic real added by B ∩ E evades all Borel null sets with Borel code in E. (3) The generic real added by D ∩ E dominates all the functions in E. (4) The generic real added by E ∩ E is eventually different from all the functions in E.

c pre =θ 0 covFigure 6 .
Figure 6.The cardinals ϑ n and θ n are increasing along the arrows.The upper diagram shows the situation forced by P 0 , and the lower diagram shows the one forced by P 1 .(s can be anywhere between p and b.) 1 and p 2 forcing the same, but this contradicts (c) because c 1 , c 2 ∈ F p and c 1 R e c 2 .(e) is straightforward.□ Remark 4.3.The obvious restriction of G B to, say, the first two coordinates, is not a projection, and G B is not a FS iteration of length ω 1 in any natural way.Assume, e.g., we restrict to {0, 1} ⊆ B = ω 1 , and B contains an e-colored edge from node e to node 2 for e ∈ {0, 1}.Start with a condition p : {0, 1, 2} × n → 2 limit then P * γ = ξ<γ P * ξ .Denote P * := P * π .Note that tidy Sλs iterations are coherent in the sense that P * η ∩ P ξ = P * ξ for any π 1 ≤ ξ ≤ η ≤ π.Conditions (5) and (6) were included to guarantee this.Definition 5.4.Let P be a tidy Sλs iteration.(1)Forπ 1 ≤ ξ ≤ π define the refined history domainξ + = ξ ∪ ζ∈ξ∩Π {ζ} × Ξ ζ .(2)For p ∈ P * and a P * -name τ we define the refined history H * (p) ⊆ π + and H * (τ ) ⊆ π + as follows.For π 1 ≤ ξ ≤ π we define H * by recursion on ξ for p ∈ P * ξ and for a P * ξ -name τ .(i) For p ∈ P * π1 , H * (p) := H(p).(ii) For ξ = ζ +1 and p ∈ P * ζ+1 , H * (p) = H * (p↾ζ) when ζ / ∈ supp p, otherwise: , where suitable now means the following (recall that we have Qζ = E P * ζ ↾C ζ ): For any sequence of nice P * ζ ↾C ζ -names for elements of E, 16 the D ε α -limit of this sequence is forced to be in Qζ as well.Here, we only get a λ-club and not an ω 1 -club, as only for increasing unions of length λ we have