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# Soft-linkedness

Miguel A. Cardona  Institute of Mathematics, Pavol Jozef Šafárik University,
041 80, Šrobárova 1014/2, 040 01 Košice, Slovakia<EMAIL_ADDRESS>https://www.researchgate.net/profile/Miguel_Cadona_Montoya

###### Abstract.

We have revised the softness property introduced by Brendle and Judah,
resulting in the definition of a class of posets called $\sigma$-soft-linked.
Our research demonstrates that these posets work well to preserve the evasion
number as well as the bounding number small in generic extensions.
Furthermore, we establish a connection between our concept and the Fréchet-
linked notion introduced by Mejía.

###### Key words and phrases:

Soft-linked, Frechet-linked, evasion number, preservation of evasion number,
Cichoń’s diagram

###### 2020 Mathematics Subject Classification:

03E05,03E17,03E35,03E40

The author was supported by the Slovak Research and Development Agency under
Contract No. APVV-20-0045 and by Pavol Jozef Šafárik University at a
postdoctoral position.

##  1\. Introduction

Blass’s [Bla94] study of set-theoretic aspects of the Specker phenomenon in
Abelian group theory led him to introduce the concept of evasion and
prediction, which is a combinatorial concept. He investigated several evasion
numbers and how large they are compared to other cardinal characteristics.
Particularly, concerning cardinals in Cichoń’s diagram. Brendle [Bre95]
extended these ideas and studied them more closely.

Fix $b\in((\omega+1)\smallsetminus 2)^{\omega}$. An _$\prod b$ -predictor_ is
a pair $\pi=(D,\langle\pi_{n}\mid n\in D\rangle)$ such that
$D\in[\omega]^{\aleph_{0}}$ and $\pi_{n}:\prod_{k<n}b(k)\to b(n)$ for $n\in
D$. Let $\Pi_{b}$ be the collection of $\prod b$-predictors.

For $\pi\in\Pi_{b}$ and $f\in\prod b$ define

$\pi\emph{\ predicts\ }f\text{\ iff \
}(f(n)=\pi_{n}(f{\upharpoonright}n))\text{\ for all but finitely many\ }n\in
D;$

otherwise _$f$ evades $\pi$_. We define the corresponding  _evasion number_

$\mathfrak{e}_{b}:=\mathfrak{e}_{\prod b}=\min\\{|F|\mid\,F\subseteq\prod
b\textrm{\ and\ }\forall\pi\in\Pi_{b}\,\exists f\in F\,(f\text{\ evades\
}\pi)\\}$

It is clear that $f\leq g$ we have that
$\mathfrak{e}_{g}\leq\mathfrak{e}_{f}$. In the case that $b$ eventually equals
$\omega$, we get $\mathfrak{e}:=\mathfrak{e}_{b}$ the _evasion number_ ; in
case that $\prod b=n^{\omega}$ ($n\geq 2$), we set
$\mathfrak{e}_{n}:=\mathfrak{e}_{b}$. Finally, we let
$\mathfrak{e}_{\mathbf{ubd}}:=\min\\{\mathfrak{e}_{b}\mid\,b\in\omega^{\omega}\\}$
the _unbounded evasion number_ and let
$\mathfrak{e}_{\textbf{fin}}:=\min\\{\mathfrak{e}_{n}\mid\,n\in\omega\\}$ the
_finite evasion number_. It is well-known that
$\mathfrak{e}\leq\mathfrak{e}_{\mathbf{ubd}}\leq\mathfrak{e}_{\mathbf{fin}}$.
Even more, it is not hard to see that
$\mathfrak{e}_{\mathbf{fin}}\leq\mbox{\rm non}(\mathcal{E})$. Below we
summarize some ZFC results related to the evasion numbers.

######  Lemma 1.1 ([Bre95, Sec. 4.2]).

  1. (1)

$\mathfrak{e}\geq\min\\{\mathfrak{b},\mathfrak{e}_{\mathbf{ubd}}\\}$.
Moreover,
$\min\\{\mathfrak{b},\mathfrak{e}\\}=\min\\{\mathfrak{b},\mathfrak{e}_{\mathbf{ubd}}\\}$,
so $\mathfrak{b}>\mathfrak{e}$ implies
$\mathfrak{e}=\mathfrak{e}_{\mathbf{ubd}}$

  2. (2)

$\mathfrak{e}_{\mathbf{fin}}=\mathfrak{e}_{n}$ for all $n\geq 2$.

  3. (3)

$\mathfrak{e}_{\mathbf{fin}}\geq\mathfrak{s}$, here $\mathfrak{s}$ donotes the
_splitting number_.

  4. (4)

$\mathfrak{e}\leq\mbox{\rm cov}(\mathcal{M})\leq\mathfrak{d}$.

  5. (5)

$\mathfrak{e}_{\mathbf{fin}}\leq\mbox{\rm non}(\mathcal{E})$.

  6. (6)

$\mbox{\rm add}(\mathcal{N})\leq\mathfrak{e}$ and
$\min\\{\mathfrak{e},\mathfrak{b}\\}\leq\mbox{\rm add}(\mathcal{M})$.

Here, given an ideal $\mathcal{I}\subseteq\mathcal{P}(X)$ containing all
singletons, let

$\displaystyle\mbox{\rm add}(\mathcal{I})$
$\displaystyle=\min\\{|\mathcal{J}|\mid\,\mathcal{J}\subseteq\mathcal{I},\,\bigcup\mathcal{J}\notin\mathcal{I}\\};$
$\displaystyle\mbox{\rm cov}(\mathcal{I})$
$\displaystyle=\min\\{|\mathcal{J}|\mid\,\mathcal{J}\subseteq\mathcal{I},\,\bigcup\mathcal{J}=X\\};$
$\displaystyle\mbox{\rm non}(\mathcal{I})$
$\displaystyle=\min\\{|A|\mid\,A\subseteq X,\,A\notin\mathcal{I}\\};$
$\displaystyle\mbox{\rm cof}(\mathcal{I})$
$\displaystyle=\min\\{|\mathcal{J}|\mid\,\mathcal{J}\subseteq\mathcal{I},\
\forall\,A\in\mathcal{I}\ \exists\,B\in\mathcal{J}\,(A\subseteq B)\\}.$

$\mathfrak{d}$ denotes as usual the _dominating number_ , that is, the size of
the smallest family of functions $D\subseteq\omega^{\omega}$ such that for all
$f\in\omega^{\omega}$ there is $g\in D$ with $g(n)>f(n)$ for almost all $n$.
In future, we shall abbreviate _for almost all $n$_ by $\forall^{\infty}n$,
and write $f\leq^{*}g$ for $\forall^{\infty}n(f(n)\leq g(n))$. Call a family
of functions $F\subseteq\omega^{\omega}$ unbounded if for all
$f\in\omega^{\omega}$ there is $g\in F$ which lies infinitely often above $f$.
Then

$\mathfrak{b}=\min\\{F\subseteq\omega^{\omega}\mid\,F\text{\ is unbounded}\\}$

is the _unbounding number_ which is dual to $\mathfrak{d}$. Finally, denote by
$\mathcal{N}$ and $\mathcal{M}$ the $\sigma$-ideals of Lebesgue null sets and
of meager sets of $2^{\omega}$, respectively, and let $\mathcal{E}$ be the
$\sigma$-ideal generated by closed measure zero subsets of $2^{\omega}$.

We complete this cycle of ZFC results with another result of Brendle
concerning the relationship between $\mathfrak{e}_{\mathbf{ubd}}$ and
$\mbox{\rm non}(\mathcal{N}\\!\mathcal{A})$ where $\mathcal{N}\\!\mathcal{A}$
denotes the $\sigma$-ideal of the null-additive subsets of $2^{\omega}$ (see
2.2). Namely, he states $\mbox{\rm
non}(\mathcal{N}\\!\mathcal{A})\leq\mathfrak{e}_{\mathbf{ubd}}$, but the proof
does not appear anywhere. For completeness, we offer a proof of this
inequality in Section 2

As to consistency results, in [BJ93], Brendle and Judad presented the property
that helps us force $\mathfrak{b}$ small. Given partial order $\mathds{P}$, a
function $h\colon\mathds{P}\to\omega$ is a _height function_ iff $q\leq p$
implies $h(q)\geq h(p)$. A pair $\langle\mathds{P},h\rangle$ has the property
() iff:

() given a maximal antichain $\\{p_{n}\mid\,n\in\omega\\}\subseteq\mathds{P}$
and $m\in\omega$, there is a $n\in\omega$ such that: whenever $p$ is
incompatible with $\\{p_{j}\mid\,j\in n\\}$ then $h(p)>m$.

They established that FS (finite support) iterations of ccc posets that
fulfill () do not add dominating reals (so they do not increase $\mathfrak{b}$
small). Later, Brendle [Bre95] enhanced () to prove the consistency of
$\mathfrak{e}<\mathfrak{e}_{\mathbf{ubd}}$. He showed that FS iterations of
ccc posets with the property () do not add predictors (so they do not increase
$\mathfrak{e}$) as well as do not add dominating reals. We say that a pair
$\langle\mathds{P},h\rangle$ has the property () iff:

() (I) given a $p\in\mathds{P}$, a maximal antichain
$\\{p_{n}\mid\,n\in\omega\\}\subseteq\mathds{P}$ of conditions below $p$ and
$m\in\omega$, there is a $n\in\omega$ such that: whenever $q\leq p$ is
incompatible with $\\{p_{j}\mid\,j\in n\\}$ then $h(p)>m$; and (II) if
$p,q\in\mathds{P}$ are compatible, then there is $r\leq p,q$ so that $h(r)\leq
h(p)+h(q)$.

Drawing inspiration from the property labeled as (), the objective of this
note is to establish a class of partially ordered sets (posets) referred to as
_$\sigma$ -soft-linked_ (3.1). This class encompasses widely recognized types
of forcing notions such as eventually different real forcing, meager forcing,
Dirichlet–Minkowski forcing, and others. Furthermore, we will provide evidence
that any poset in this category, which is definable, maintains a small value
of $\mathfrak{e}$. Additionally, we will demonstrate that our notion is
connected to the concept of ”Fréchet-linked in $\mathds{P}$” (abbreviated as
”$\mathrm{Fr}$-linked”) as introduced by Mejía [Mej19].

Notation. A  _forcing notion_ is a pair $\langle\mathds{P},\leq\rangle$ where
$\mathds{P}\neq\emptyset$ and $\leq$ is a relation on $\mathds{P}$ that
satisfies reflexivity and transitivity. We also use the expression pre-ordered
set (abbreviated p.o. set or just poset) to refer to a forcing notion. The
elements of $\mathds{P}$ are called conditions and we say that a condition $q$
is _stronger_ than a condition $p$ if $q\leq p$.

######  Definition 1.2.

Let $\mathds{P}$ be a forcing notion.

  1. (1)

Say that $p,q\in\mathds{P}$ are _compatible_ (in $\mathds{P}$), denoted by
$p\,\|_{\mathds{P}}\,q$, if $\exists r\in\mathds{P}\,(r\leq p\textrm{\ and\
}r\leq q)$. Say that $p,q\in\mathds{P}$ are _incompatible_ (in $\mathds{P}$)
if they are not compatible in $\mathds{P}$, which is denoted by
$p\perp_{\mathds{P}}q$.

When $\mathds{P}$ is clear by the context, we just write $p\,\|\,q$ and
$p\perp q$.

  2. (2)

Say that $A\subseteq\mathds{P}$ is an _antichain_ if $p,q\in\mathds{P}\,(p\neq
q\Rightarrow p\perp q)$. $A$ is a _maximal antichain_ on $\mathds{P}$ iff $A$
is an antichain and $p\in\mathds{P}\,\exists q\in A\,(p\,\|_{\mathds{P}}\,q)$.

  3. (3)

Say that $D\subseteq\mathds{P}$ is _dense_ (in $\mathds{P}$) if
$p\in\mathds{P}\,\exists q\in D\,(q\leq p)$.

  4. (4)

Say that $G\subseteq\mathds{P}$ is a _$\mathds{P}$ -filter_ if it satisfies

    1. (a)

$G\neq\emptyset$;

    2. (b)

for all $p,q\in G$ there is some $r\in G$ such that $r\leq p$ and $r\leq p$;
and

    3. (c)

if $p\in\mathds{P}$, $q\in G$ and $q\leq p$, then $p\in G$.

  5. (5)

Let $\mathcal{D}$ be a family of dense subsets of $\mathds{P}$. Say that
$G\subseteq\mathds{P}$ is _$\mathds{P}$ -generic over $D$_ if $G$ is a
$\mathds{P}$-filter and $D\in\mathcal{D}\,(G\cap D\neq\emptyset$. Denote by
$\dot{G}_{\mathds{P}}$ the canonical name of the generic set. When
$\mathds{P}$ is clear by the context, we just write $\dot{G}$.

######  Fact 1.3 ([Gol93]).

Let $\mathds{P}$ be a forcing notion. Let $p,q\in\mathds{P}$.

  1. (1)

$p\perp q$ iff $q\Vdash p\notin\dot{G}$.

  2. (2)

$G\subseteq\mathds{P}$ is a $\mathds{P}$-generic over $V$ iff for every
maximal antichain $A\in V$, $|G\cap A|=1$.

Define the order $\leq^{*}$ in $\mathds{P}$ as $q\leq^{*}p$ iff $\forall r\leq
q\,(r\parallel p)$, i.e, $p\leq^{*}q\Leftrightarrow\\{q\\}$ is predense below
$p$. $\mathds{P}$ is separative if and only if ${\leq^{*}}$ equals ${\leq}$.
For $p\in\mathds{P}$ let $\mathds{P}|p=\\{q\in\mathds{P}\mid\,q\leq p\\}$.

We now review some basic notation about relational systems. A  _relational
system_ is a triple $\mathbf{R}=\langle X,Y,R\rangle$ where $R$ is a relation
and $X$ and $Y$ are non-empty sets. Such a relational system has two cardinal
characteristics associated with it:

$\displaystyle\mathfrak{b}(\mathbf{R})$
$\displaystyle:=\min\\{|F|\mid\,F\subseteq X\text{\ and }\neg\exists y\in
Y\,\forall x\in F\colon xRy\\},$ $\displaystyle\mathfrak{d}(\mathbf{R})$
$\displaystyle:=\min\\{|D|\mid\,D\subseteq Y\text{\ and }\forall x\in
X\,\exists y\in D\colon xRy\\}.$

Given another relational system $\mathbf{R}^{\prime}=\langle
X^{\prime},Y^{\prime},R^{\prime}\rangle$, say that a pair
$(\Psi_{-},\Psi_{+})\colon\mathbf{R}\to\mathbf{R}^{\prime}$ is a _Tukey
connection from $\mathbf{R}$ into $\mathbf{R}^{\prime}$_ if $\Psi_{-}\colon
X\to X^{\prime}$ and $\Psi_{+}\colon Y^{\prime}\to Y$ are functions such that
$\forall\,x\in X\ \forall\,y^{\prime}\in
Y^{\prime}\colon\Psi_{-}(x)R^{\prime}y^{\prime}$ then
$xR\Psi_{+}(y^{\prime})$. Say that $\mathbf{R}$ is _Tukey below_
$\mathbf{R}^{\prime}$, denoted by
$\mathbf{R}\preceq_{\mathrm{T}}\mathbf{R}^{\prime}$, if there is a Tukey
connection from $\mathbf{R}$ to $\mathbf{R}^{\prime}$. Say that $\mathbf{R}$
is _Tukey equivalence_ to $\mathbf{R}^{\prime}$, denoted by
$\mathbf{R}\cong_{\mathrm{T}}\mathbf{R}^{\prime}$ if
$\mathbf{R}\preceq_{\mathrm{T}}\mathbf{R}^{\prime}$ and
$\mathbf{R}^{\prime}\preceq_{\mathrm{T}}\mathbf{R}$. It is well-known that
$\mathbf{R}\preceq_{\mathrm{T}}\mathbf{R}^{\prime}$ implies
$\mathfrak{b}(\mathbf{R}^{\prime})\leq\mathfrak{b}(\mathbf{R})$ and
$\mathfrak{d}(\mathbf{R})\leq\mathfrak{d}(\mathbf{R}^{\prime})$. Hence,
$\mathbf{R}\cong_{\mathrm{T}}\mathbf{R}^{\prime}$ implies
$\mathfrak{b}(\mathbf{R}^{\prime})=\mathfrak{b}(\mathbf{R})$ and
$\mathfrak{d}(\mathbf{R})=\mathfrak{d}(\mathbf{R}^{\prime})$.

For instance, the cardinal characteristics associated with an ideal can be
characterized by relational systems.

######  Example 1.4.

For $\mathcal{I}\subseteq\mathcal{P}(X)$, define the relational systems:

  1. (1)

$\mathcal{I}:=\langle\mathcal{I},\mathcal{I},\subseteq\rangle$, which is a
directed preorder when $\mathcal{I}$ is closed under unions (e.g. an ideal).

  2. (2)

$\mathbf{C}_{\mathcal{I}}:=\langle X,\mathcal{I},\in\rangle$.

It is well-known that, whenever $\mathcal{I}$ is an ideal on $X$ containing
$[X]^{<\aleph_{0}}$,

  1. (a)

$\mathfrak{b}(\mathcal{I})=\mbox{\rm add}(\mathcal{I})$.

  2. (b)

$\mathfrak{d}(\mathcal{I})=\mbox{\rm cof}(\mathcal{I})$.

  3. (c)

$\mathfrak{b}(\mathbf{C}_{\mathcal{I}})=\mbox{\rm non}(\mathcal{I})$.

  4. (d)

$\mathfrak{d}(\mathbf{C}_{\mathcal{I}})=\mbox{\rm cov}(\mathcal{I})$.

Here, as usual, given a formula $\phi$, $\forall^{\infty}\,n<\omega\colon\phi$
means that all but finitely many natural numbers satisfy $\phi$;
$\exists^{\infty}\,n<\omega\colon\phi$ means that infinitely many natural
numbers satisfy $\phi$.

Lastly, for $x\in\mathds{R}$ let $\parallel x\parallel$ denote the distance of
$x$ to the nearlest integer.

##  2\. Evasion number and localization

This section aims to prove the following theorem:

######  Theorem 2.1 ([Bre95]).

$\mbox{\rm non}(\mathcal{N}\\!\mathcal{A})\leq\mathfrak{e}_{\mathbf{ubd}}$.

Before entering into the details of the proof, we review some basic notions
and notation:

######  Definition 2.2.

A set $X\subseteq 2^{\omega}$ is termed _$\mathcal{N}$ -additive_ if, for
every $A\in\mathcal{N}$, $A+X\in\mathcal{N}$. Denote by
$\mathcal{N}\\!\mathcal{A}$ the collection of the $\mathcal{N}$-additive
subsets of $2^{\omega}$. Notice that $\mathcal{N}\\!\mathcal{A}$ is a
$\sigma$-ideal and $\mathcal{N}\\!\mathcal{A}\subseteq\mathcal{N}$.

We now introduce some terminology associated with slaloms.

######  Definition 2.3.

Given a sequence of non-empty sets $b=\langle b(n)\mid n\in\omega\rangle$ and
$h\colon\omega\to\omega$, define

$\displaystyle\prod b$ $\displaystyle:=\prod_{n\in\omega}b(n)$
$\displaystyle\mathcal{S}(b,h)$ $\displaystyle:=\prod_{n\in\omega}[b(n)]^{\leq
h(n)}$

For two functions $x\in\prod b$, $\varphi\in\mathcal{S}(b,h)$ and
$w\in[\omega]^{\aleph_{0}}$, we define

  1. (1)

$x\,\in^{*}\varphi\textrm{\ iff\
}\forall^{\infty}n\in\omega\,(x(n)\in\varphi(n))$.

  2. (2)

$x\in^{\circ}(\varphi,w)$ iff $\forall^{\infty}n\in w\,(x(n)\in\varphi(n))$.

We define the _localization cardinal_

$\mathfrak{b}^{\mathrm{Lc}}_{b,h}=\min\\{|F|\mid\,F\subseteq\prod
b\;\&\;\neg\exists\varphi\in\mathcal{S}(b,h)\,\forall x\in
F\,(x\in^{*}\varphi)\\},$

and a _variant of the localization cardinal_

$\mathfrak{b}^{\mathrm{Lc_{0}}}_{b,h}=\min\\{|F|\mid\,F\subseteq\prod
b\;\&\;\neg\exists(\varphi,w)\in\mathcal{S}(b,h)\times[\omega]^{\aleph_{0}}\,\forall
x\in F\,(x\in^{\circ}(\varphi,w))\\}.$

It is not hard to see that
$\mathfrak{b}^{\mathrm{Lc}}_{b,h}\leq\mathfrak{b}^{\mathrm{Lc_{0}}}_{b,h}$
holds. Let
$\mathrm{minLc}:=\min\\{\mathfrak{b}^{\mathrm{Lc}}_{b,\mathrm{id}_{\omega}}\mid\,b\in\omega^{\omega}\\}$
where $\mathrm{id}_{\omega}$ denotes the identity function on $\omega$.

Pawlikowski provided a characterization of the uniformity of
$\mathcal{N}\\!\mathcal{A}$ in terms of localization cardinals. Namely, he
proved:

######  Theorem 2.4 ([Paw85]).

$\mbox{\rm non}(\mathcal{N}\\!\mathcal{A})=\mathrm{minLc}$.

On the other hand, it was proved that
$\mathrm{minLc}=\min\\{\mathfrak{b}^{\mathrm{Lc}}_{b,h}\mid\,b\in\omega^{\omega}\\}$
when $h$ goes to infinity in [CM19, Lemma 3.8]. Therefore, $\mbox{\rm
non}(\mathcal{N}\\!\mathcal{A})=\min\\{\mathfrak{b}^{\mathrm{Lc}}_{b,h}\mid\,b\in\omega^{\omega}\\}$.

######  Definition 2.5.

Let $b,h$ as in 2.3.

  1. (1)

Define $\mathbf{Lc}(b,h):=\langle\prod b,\mathcal{S}(b,h),\in^{*}\rangle$
($\mathbf{Lc}$ stands for _localization_), which is a relational system. Put
$\mathfrak{b}^{\mathrm{Lc}}_{b,h}:=\mathfrak{b}(\mathbf{Lc}(b,h))$ and
$\mathfrak{d}^{\mathrm{Lc}}_{b,h}:=\mathfrak{d}(\mathbf{Lc}(b,h))$.

  2. (2)

Define $\mathbf{Lc}_{0}(b,h):=\langle\prod
b,\mathcal{S}(b,h)\times[\omega]^{\aleph_{0}},\in^{\circ}\rangle$, which is a
relational system. Put
$\mathfrak{b}^{\mathrm{Lc_{0}}}_{b,h}:=\mathfrak{b}(\mathbf{Lc}_{0}(b,h))$ and
$\mathfrak{d}^{\mathrm{Lc_{0}}}_{b,h}:=\mathfrak{d}(\mathbf{Lc}_{0}(b,h))$.

  3. (3)

Assume $b\in((\omega+1)\smallsetminus 2)^{\omega}$. Define
$\mathbf{E}_{b}=\langle\prod b,\Pi_{b},=^{\textbf{pr}}\rangle$
($\mathbf{E}_{b}$ stands for _unbounded evasion_) where $=^{\textbf{pr}}$
denotes the relation introduced in the definition of evasion number, which is
a relational system. So it is clear that
$\mathfrak{b}(\mathbf{E}_{b})=\mathfrak{e}_{b}$ and
$\mathfrak{d}(\mathbf{E}_{b})=\mathfrak{pr}_{b}$ ($\mathfrak{pr}_{b}$ is dual
to $\mathfrak{e}_{b}$ in a natural sense).

######  Remark 2.6.

Notice that, for fixed $\pi\in\Pi_{b}$, $\\{f\in\prod
b\mid\,f\in^{\textbf{pr}}\pi\\}$ is meager set whenever
$b\in((\omega+1)\smallsetminus 2)^{\omega}$, so
$\mathbf{C}_{\mathcal{M}}\preceq_{\mathrm{T}}\mathbf{E}_{b}$, which implies
that $\mathfrak{b}(\mathbf{E}_{b})\leq\mbox{\rm non}(\mathcal{M})$ and
$\mbox{\rm cov}(\mathcal{M})\leq\mathfrak{d}(\mathbf{E}_{b})$.

Thanks to Theorem 2.4 to prove Theorem 2.1, it suffices to prove that
$\mathrm{minLc}\leq\mathfrak{e}_{\mathbf{ubd}}$.

###### Proof of Theorem 2.1.

Let $b\in\omega^{\omega}$. Define $c_{b}\in\omega^{\omega}$ by
$c_{b}(n)=\prod_{i\in I_{n}}b(i)$ where
$I_{n}:=[\frac{n(n+1)}{2},\frac{n(n+1)}{2}+1]$. Note that $\langle I_{n}\mid
n\in\omega\rangle$ is a partition of $\omega$ into adjacent intervals such
that the $n$-th interval $I_{n}$ has size $n+1$. Hence, it is enough to prove
that
$\mathbf{E}_{b}\preceq_{\mathrm{T}}\mathbf{Lc}_{0}(c_{b},\mathrm{id}_{\omega})$.
To see this, we have to find maps $\Psi_{-}\colon\prod b\to\prod c_{b}$ and
$\Psi_{+}\colon\mathcal{S}(c_{b},\mathrm{id}_{\omega})\to\Pi_{b}$ such that,
for any $x\in\prod b$ and for any
$\varphi\in\mathcal{S}(c_{b},\mathrm{id}_{\omega})$,
$\Psi_{-}(x)\in^{*}\varphi$ implies $x=^{\textbf{pr}}\Psi_{+}(\varphi)$.

Let $x\in F$. Define $y_{x}\in\prod c_{b}$ by
$y_{x}(n)=x{\upharpoonright}I_{n}$ for each $n\in\omega$. Put
$\Psi_{-}(x)=y_{x}$. On the other hand, for
$\varphi\in\mathcal{S}(c_{b},\mathrm{id}_{\omega})$ and for
$w\in[\omega]^{\aleph_{0}}$. We can interpret $\varphi(n)$ as a subset of the
$(n+1)$-dimensional $Q$-vector space $Q^{I_{n}}$. Since $|\varphi(n)|=n$,
there is a non-zero functional $a^{n}:Q^{I_{n}}\to Q$ which annihilates every
vector in $\varphi(n)$. For $i\in I_{n}$ let $e_{i}\in Q^{I_{n}}$ be the unit
vector defined by $e_{i}(j)=\delta_{i,j}$ (Kronecker), let
$a_{i}^{n}=a^{n}(e_{i})$ and let $d_{n}=\max\\{i\in I_{n}\mid\,a^{n}_{i}\neq
0\\}$. Then it is clear that for every $x\in F$ and $n\in w$ with
$y_{x}(n)\in\varphi(n)$, rewriting $a^{n}(y_{x}(n))=0$, we obtain

$x(d_{n})=-\frac{1}{d_{n}}\sum_{i\in I_{n}\cap d_{n}}a_{i}^{n}\cdot x(i)$

Now, define a predictor $\Psi_{+}(\varphi)=\langle\pi_{d}\mid d\in D\rangle$
by

$D:=\\{d_{n}\mid\,n\in\omega\\}\textrm{\ and\
}\pi_{d}(\sigma)=-\frac{1}{d_{n}}\sum_{i\in I_{n}\cap
d_{n}}a_{i}^{n}\cdot\sigma(I).$

It is not hard to see that $y_{x}\in^{*}\varphi$ implies
$x=^{\textbf{pr}}\pi$. ∎

##  3\. Soft-linkedness

Within this section, our primary objective is to formalize the approach of
Brendle and Judad in order to introduce the central concept of this study,
called “$\mathrm{leaf}$-linked”. Additionally, we will investigate the
correlation between $\mathrm{leaf}$-linked and $\mathrm{Fr}$-linked, as
demonstrated in Theorem 3.5.

###  3.1. Soft-linkedness

We now proceed to acquaint our linkedness property.

######  Definition 3.1.

Let $\mathds{P}$ be a forcing notion and $\theta$ be a cardinal number.

  1. (1)

A set $Q\subseteq\mathds{P}$ is _leaf-linked_ if, for any maximal antichain
$\\{p_{n}\mid\,n\in\omega\\}\subseteq\mathds{P}$, there is some $n\in\omega$
such that $\forall p\in Q\,\exists j<n\,(p\,||\,p_{j}).$

  2. (2)

A set $Q\subseteq\mathds{P}$ is _$\mathit{leaf}^{*}$ -linked_ if, for any
$p\in\mathds{P}$ and for any maximal antichain
$\\{p_{n}\mid\,n\in\omega\\}\subseteq\mathds{P}$ of conditions below $p$,
there is some $n\in\omega$ such that $\forall q\in Q\,[q\leq
p)\Rightarrow\exists j<n\,(q\,||\,p_{j})]$.

  3. (3)

Say that $\mathds{P}$ is _$\theta$ -$\mathrm{soft}$-linked_ if there is a
sequence $\langle Q_{\alpha}\mid\alpha<\theta\rangle$ of subsets of
$\mathds{P}$ fulfilling the following:

    1. (a)

$Q_{\alpha}$ is $\mathrm{leaf}$-linked,

    2. (b)

$\bigcup_{\alpha<\theta}Q_{\alpha}$ is dense in $\mathds{P}$, and

    3. (c)

$\forall\alpha,\alpha^{\prime}<\theta\,\exists\alpha^{*}<\theta\,\forall p\in
Q_{\alpha}\,\forall q\in Q_{\alpha^{\prime}}\,[p\,||q\Rightarrow\exists r\in
Q_{\alpha^{*}}(r\leq p\wedge r\leq q)]$.

######  Remark 3.2.

Let $Q\subseteq\mathds{P}$.

  1. (1)

If $Q$ is $\mathrm{leaf}$-linked, then $Q$ is $\mathrm{leaf}^{*}$-linked.

  2. (2)

$Q$ is $\mathrm{leaf}^{*}$-linked iff for all $p\in\mathds{P}$, $Q|p:=\\{q\in
Q\mid\,q\leq p\\}$ is $\mathrm{leaf}$-linked.

In the ensuing discussion, our main objective is to establish the correlation
between our concept and the $\sigma$-Fréchet-linked notion introduced by
Mejía. He demonstrated that a $\sigma$-Fr-linked poset does not add dominating
reals and that it preserves a specific type of maximal almost disjoint (mad)
families. The findings in Mejía’s work were later enhanced in the study
conducted by Brendle, the author, and Mejía [BCM21], which served as the
motivation for constructing matrix iterations of ${<}\theta$-uf-linked posets.
The purpose of these iterations was to enhance the separation of the left-hand
side of Cichoń’s diagram from the study conducted by Goldstern, Mejía, and
Shelah [GMS16], by incorporating the inequality $\mbox{\rm
cov}(\mathcal{M})<\mathfrak{d}=\mbox{\rm non}(\mathcal{N})=\mathfrak{c}$.

We begin with some notation:

  * •

Denote by $\mathrm{Fr}:=\\{\omega\smallsetminus
a\mid\,a\in[\omega]^{<\aleph_{0}}\\}$ the _Frechet filter_.

  * •

A filter $F$ on $\omega$ is _free_ if $\mathrm{Fr}\subseteq F$. A set
$x\subseteq\omega$ is _$F$ -positive_ if it intersects every member of $F$.
Denote by $F^{+}$ the family of $F$-positive sets. Note that
$x\in\mathrm{Fr}^{+}$ iff $x$ is an infinite subset of $\omega$.

######  Definition 3.3 ([Mej19, BCM21]).

Let $\mathds{P}$ be a poset and $F$ a filter on $\omega$. A set
$Q\subseteq\mathds{P}$ is _$F$ -linked_ if, for any $\bar{p}=\langle p_{n}\mid
n<\omega\rangle\in Q^{\omega}$, there is some $q\in\mathds{P}$ forcing that

$q\Vdash\\{n\in\omega\mid\,p_{n}\in\dot{G}\\}\in F^{+}.$

Observe that, in the case $F=\mathrm{Fr}$, the above equation is
“$\\{n\in\omega\mid\,p_{n}\in\dot{G}\\}$ is infinite”.

We say that $Q$ is _uf-linked (ultrafilter-linked)_ if it is $F$-linked for
any filter $F$ on $\omega$ containing the _Frechet filter_ $\mathrm{Fr}$.

For an infinite cardinal $\theta$, $\mathds{P}$ is _$\mu$ -$F$-linked_ if
$\mathds{P}=\bigcup_{\alpha<\theta}Q_{\alpha}$ for some $F$-linked
$Q_{\alpha}$ ($\alpha<\theta$). When these $Q_{\alpha}$ are uf-linked, we say
that $\mathds{P}$ is _$\mu$ -uf-linked_.

It is clear that any uf-linked set $Q\subseteq\mathds{P}$ is F-linked, which
implies $\mathrm{Fr}$-linked. But, for ccc poset we have:

######  Lemma 3.4 ([Mej19, Lem 5.5]).

If $\mathds{P}$ is ccc then any subset of $\mathds{P}$ is uf-linked iff it is
Fr-linked.

One of the main results of this section is:

######  Theorem 3.5.

Assume that $\mathds{P}$ is ccc. If $\mathds{P}$ is
$\theta$-$\mathrm{soft}$-linked poset, then it is
$\theta$-$\mathrm{Fr}$-linked.

In order to prove the theorem above, all we have to do is prove that any
subset of a poset $\mathds{P}$ is $\mathrm{Fr}$-linked, it is still leaf-
linked. However, it is worth noting that we have much more:

######  Lemma 3.6.

Let $\mathds{P}$ be a ccc forcing notion and a set $Q\subseteq\mathds{P}$. The
following statements are equivalent:

  1. (1)

$Q$ is $\mathrm{Fr}$-linked.

  2. (2)

$Q$ is $\mathrm{leaf}$-linked.

  3. (3)

for all $\mathds{P}$-name $\dot{n}$ of a natural number there is some
$m<\omega$ such that $\forall p\in Q\,(p\not\Vdash m<\dot{n})$.

###### Proof.

$\ref{Fr}\Rightarrow\ref{Soft}$: Let $\\{p_{n}\mid\,n\in\omega\\}$ be a
maximal antichain in $\mathds{P}$. Towards a contradiction let us assume
$\forall m\,\exists q_{m}\in Q\,\forall n<m\,(q_{m}\perp p_{n})$. By
$\mathrm{Fr}$-linkedness, choose $q\in\mathds{P}$ such that
$q\Vdash``\exists^{\infty}m\in\omega\,(q_{m}\in\dot{G})"$. Since each $q_{m}$
fulfills $\forall n<m\,(q_{m}\perp p_{n})$, $q\Vdash\exists^{\infty}m\forall
n<m\,(p_{n}\not\in\dot{G})$. Hence, $q\Vdash\forall
n<\omega\,(p_{n}\not\in\dot{G})$. On the other hand, since
$\\{p_{n}\mid\,n\in\omega\\}$ is a maximal antichain, by using 1.3
$\Vdash\exists n\in\omega\,(p_{n}\in\dot{G})$, which establishes a
contradiction.

$\ref{Soft}\Rightarrow\ref{ConFr}$: Assume that $\dot{n}$ is a
$\mathds{P}$-name of a natural number and assume towards a contradiction that
for each $m\in\omega$, there is some $q_{m}\in Q$ that forces $m<\dot{n}$. Let
$\\{p_{k}\mid\,k\in\omega\\}$ be a maximal antichain deciding the value
$\dot{n}$ and let $n_{k}$ be such that $p_{k}\Vdash``\dot{n}=n_{k}"$. By soft-
linkedness, choose $m\in\omega$ such that $\forall p\in Q\,\exists
k<m\,(p\parallel p_{k})$. Next, let $k^{\star}$ be larger than
$\max\\{n_{k}\mid\,k<m\\}$. Then $q_{k^{\star}}\Vdash k^{\star}<\dot{n}$ and
since $q_{k^{\star}}\in Q$, then there is $k<m$ such that
$q_{k^{\star}}\parallel p_{k}$. Let $r\leq q_{k^{\star}},p_{k}$. Then
$r\Vdash\dot{n}=k<k^{\star}<\dot{n}$, which is a contradiction.

$\neg\ref{Fr}\Rightarrow\neg\ref{ConFr}$: Assume that there is a sequence
$\langle p_{n}\mid n<\omega\rangle$ in $Q$ such that
$\Vdash``\\{n\in\omega\mid\,p_{n}\in\dot{G}\\}$ finite”. Then there is a
$\mathds{P}$-name $\dot{M}$ in $\omega$ such that
$\Vdash\\{n\in\omega\mid\,p_{n}\in\dot{G}\\}\subseteq\dot{M}$. Towards a
contradiction suppose that $\exists m\,\forall q\in Q\,(q\not\Vdash
m<\dot{M})$. Choose $m\in\omega$ such that $\forall q\in Q\,(q\not\Vdash
m<\dot{M})$. For any $n\in\omega$, $p_{n}\not\Vdash m<\dot{M}$. Next, find
$r\leq p_{m}$ such that $r\Vdash m\geq\dot{M}$, on the other hand, since
$r\Vdash p_{m}\in G$, $r\Vdash m<\dot{M}$. This is a contradiction. ∎

######  Remark 3.7.

We, along with Mejía, proved the direction $\ref{ConFr}\Rightarrow\ref{Fr}$ in
3.6.

Next, we proceed to derive several ramifications stemming from 3.1. These
ramifications hold significant relevance in substantiating the theorem about
the preservation of the evasion number, as explored further in Theorem 4.19.

######  Lemma 3.8.

Let $\theta$ be a cardinal number. Consider the following properties of a
poset $\mathds{P}$:

  1. (1)

$\theta$-soft-linked.

  2. (2)

There is a sequence $\langle Q_{\alpha}\mid\alpha<\theta\rangle$ of soft-
linked subsets of $\mathds{P}$ such that $Q=\bigcup_{\alpha<\theta}Q_{\alpha}$
is dense in $\mathds{P}$ and there are functions $h\colon Q\to\theta$ and
$g^{\star}\colon\theta\times Q\to\theta$ such that $\forall\alpha<\theta$
$h^{-1}\llbracket\\{\alpha\\}\rrbracket\subseteq Q_{\alpha}$ and

(2) $\forall\alpha<\theta\,\forall p\in Q_{\alpha}\,\forall p^{\prime}\in
Q\,[p\parallel p^{\prime}\Rightarrow\exists r\in
Q_{g(\alpha,p^{\prime})}\,(r\leq p\text{ and }r\leq p^{\prime})].$

  3. (3)

There is a sequence $\langle Q_{\alpha}\mid\alpha<\theta\rangle$ of soft-
linked subsets of $\mathds{P}$ such that $Q=\bigcup_{\alpha<\theta}Q_{\alpha}$
is dense in $\mathds{P}$ and there is $\bar{g}\colon Q\times Q\to\theta$ such
that

(3) $\forall p,p^{\prime}\in Q\,[p\parallel p^{\prime}\Rightarrow\exists r\in
Q_{\bar{g}(p,p^{\prime})}\,(r\leq p\text{ and }r\leq p^{\prime})].$

Then
$\textup{\ref{solfch:A}}\Rightarrow\textup{\ref{solfch:B}}\Rightarrow\textup{\ref{solfch:C}}$.

###### Proof.

In all cases, we can define $h\colon Q\to\theta$ by

$h(p)=\min\\{\alpha<\theta\mid\,p\in Q_{\alpha}\\}$

for $p\in Q$ because $Q=\bigcup_{\alpha<\theta}Q_{\alpha}$.

$\ref{solfch:A}\Rightarrow\ref{solfch:B}$. By emplying 3.1 (3) (3)(c), we can
define a function $g\colon\theta\times\theta\to\theta$ such that
$g\colon(\alpha,\alpha^{\prime})\mapsto\alpha^{*}$. Define
$g^{\star}\colon\theta\times Q\to\theta$ by
$g^{\star}(\alpha,p)=g(\alpha,h(p))$ for $(\alpha,p)\in\theta\times Q$. Assume
that $p\in Q_{\alpha}$ and $p^{\prime}\in Q$ and $p\parallel p^{\prime}$. Then
$p^{\prime}\in Q_{h(p^{\prime})}$ and
$g^{\star}(\alpha,p^{\prime})=g(\alpha,h(p^{\prime}))$. Therefore by (1) there
is $r\in Q_{g(\alpha,h(p^{\prime}))}=Q_{g^{\star}(\alpha,p^{\prime})}$ such
that $r\leq p,p^{\prime}$.

$\ref{solfch:B}\Rightarrow\ref{solfch:C}$. Define $G\colon Q\times Q\to\theta$
by $\bar{g}(p,p^{\prime})=g^{\star}(h(p),p^{\prime})$ for $(p,p^{\prime})\in
Q\times Q$. Assume that $p,p^{\prime}\in Q$ and $p\parallel p^{\prime}$. Then
$p\in Q_{h(p)}$ and by (2) there is $r\in
Q_{g^{\star}(h(p),p^{\prime})}=Q_{\bar{g}(p,p^{\prime})}$ such that $r\leq
p,p^{\prime}$. ∎

The following lemma establishes two more consequences of $\theta$-soft-linked
and gives a characterization of $\sigma$-soft-linked in the particular case
$\theta=\omega$.

######  Lemma 3.9.

Let $\theta$ be a cardinal number. For a poset $\mathds{P}$ consider the
following properties:

  1. (1)

$\mathds{P}$ is $\theta$-soft-linked.

  2. (2)

There are a dense set $Q\subseteq\mathds{P}$ and functions $h\colon
Q\to\theta$ and $g\colon\theta\times\theta\to\theta$ with the following
properties:

    1. (i)

For every $\alpha<\theta$ the set $\\{p\in Q\mid h(p)=\alpha\\}$ is leaf-
linked.

    2. (ii)

$\forall p,q\in Q$ $[p\parallel q\Rightarrow\exists r\in Q$ $(r\leq p,q$ and
$h(r)\leq g(h(p),h(q)))]$.

  3. (3)

There are a dense set $Q\subseteq\mathds{P}$ and functions $h\colon
Q\to\theta$ and $g\colon\theta\times\theta\to\theta$ with the following
properties:

    1. (i)

For every $\alpha<\theta$ the set $\\{p\in Q\mid h(p)\leq\alpha\\}$ is leaf-
linked.

    2. (ii)

$\forall p,q\in Q$ $[p\parallel q\Rightarrow\exists r\in Q$ $(r\leq p,q$ and
$h(r)\leq g(h(p),h(q)))]$.

Then $\ref{solft:onec}\Rightarrow\ref{solft:oneb}$;
$\ref{solft:onea}\Rightarrow\ref{solft:oneb}$;
$\ref{solft:onec}\Rightarrow\ref{solft:onea}$ for regular $\theta$; and
$\ref{solft:onea}\Leftrightarrow\ref{solft:oneb}\Leftrightarrow\ref{solft:onec}$
for $\theta=\omega$.

######  Remark 3.10.

The function $h$ in conditions (2) and (3) of 3.9 may be not monotone.

###### Proof of 3.9.

The implication $\ref{solft:onec}\Rightarrow\ref{solft:oneb}$ is trivial.

$\ref{solft:onea}\Rightarrow\ref{solft:oneb}$. Let $\langle
Q_{\alpha}\mid\alpha<\theta\rangle$ be a sequence of subsets of $\mathds{P}$
witnessing that $\mathds{P}$ is $\theta$-soft-linked. The set
$Q:=\bigcup_{\alpha<\theta}Q_{\alpha}$ is a dense subset of $\mathds{P}$. For
$p\in Q$ define $h(p)=\min\\{\alpha<\theta\mid p\in Q_{\alpha}\\}$. By using
3.1 (3) (3)(c) for every $\alpha,\beta<\theta$ define

$g(\alpha,\beta)=\min\\{\gamma<\theta\mid\,\forall p\in Q_{\alpha}\forall q\in
Q_{\beta}\,[p\parallel q\Rightarrow\exists r\in Q_{\gamma}\,(r\leq p,q)]\\}.$

Condition (i) holds because every set $\\{p\in Q\mid h(p)=\alpha\\}$ is a
subset of the leaf-soft set $Q_{\alpha}$ and is therefore leaf-linked. Assume
$p,q\in Q$ and $p\parallel q$. Then $p\in Q_{h(p)}$ and $q\in Q_{h(q)}$ and so
there is $r\in Q_{g(h(p),h(q))}$ such that $r\leq p$ and $r\leq q$. By
definition of $h$, $h(r)\leq g(h(p),h(q))$ and hence (ii) holds.

$\ref{solft:onec}\Rightarrow\ref{solft:onea}$ for regular $\theta$. By using
(i) for every $\alpha<\theta$ the set $Q_{\alpha}=\\{p\in
Q\mid\,h(p)\leq\alpha\\}$ is leaf-soft and
$Q=\bigcup_{\alpha<\theta}Q_{\alpha}$ is a dense subset of $\mathds{P}$. We
verify the condition (3)(c) in 3.1 (3). Fix
$\alpha,\alpha^{\prime}\leq\theta$. Define

$\alpha^{*}=\sup\\{g(\xi,\eta)\mid\,\xi,\eta\leq\max\\{\alpha,\alpha^{\prime}\\}\\};$

$\alpha^{*}<\theta$ because $\theta$ is regular. Assume $p\in Q_{\alpha}$,
$q\in Q_{\alpha^{\prime}}$, and $p\parallel q$. Then
$g(h(p),h(q))\leq\alpha^{*}$ because $h(p)\leq\alpha$ and
$h(q)\leq\alpha^{\prime}$. By using (ii) there is $r\leq p,q$ such that
$h(r)\leq g(h(p),h(q))\leq\alpha^{*}$ and hence $r\in Q_{\alpha^{*}}$.

$\ref{solft:oneb}\Rightarrow\ref{solft:onec}$ for $\theta=\omega$. Note that
$\\{p\in Q\mid\,h(p)\leq n\\}$ is a finite union of leaf-soft sets $\\{p\in
Q\mid\,h(p)=k\\}$ for $k\leq n$ and is therefore leaf-soft. ∎

In the rest of this section, we study some properties of pairs
$\langle\mathds{P},h\rangle$ which allows us to guarantee that $\mathds{P}$ is
$\sigma$-soft-linked. We first fix some terminology.

Let $P,Q\subseteq\mathds{P}$ and $p\in\mathds{P}$. We write $p\perp q$ if
$p\perp q$ for all $q\in P$ and $Q\subseteq\mathds{P}$ if $p\in\mathds{Q}$ for
every $p\in P$. We consider the following property of pairs
$\langle\mathds{P},h\rangle$ where $\mathds{P}$ is a poset and and $h$ a
height function on $\mathds{P}$: Say that $\langle\mathds{P},h\rangle$ has
_flash property_ if:

() (I) given $m\in\omega$ and a finite non-maximal non-empty antichain
$P\subseteq\mathds{P}$, there is some finite subset $R$ of $\mathds{P}$ such
that $R\perp P$ and $\forall p\in\mathds{P}$ $[(p\perp P$ and $h(p)\leq
m)\Rightarrow\exists r\in R\,(p\leq r)]$; and (II) if $p,q\in\mathds{P}$ are
compatible, then there is $r\leq p,q$ so that $h(r)\leq h(p)+h(q)$.

######  Lemma 3.11.

A ccc poset $\mathds{P}$ with a height function $h\colon\mathds{P}\to\omega$
satisfying  _()_ is $\sigma$-soft-linked.

###### Proof.

For $m\in\omega$ let

$Q_{m}:=\\{p\in\mathds{P}\mid\,h(p)\leq m\\}\textrm{\ and\
}Q:=\bigcup_{m\in\omega}Q_{m}.$

To prove the lemma, it suffices to verify property (3) in 3.9 for
$\theta=\omega$, for the height function $h$, and for the function
$g(n,m)=n+m$. Condition (I) is fulfilled by property (II). To prove condition
(i) we must prove that every set $Q_{m}$ is leaf-soft. Let
$P=\\{p_{k}\mid\,k\in\omega\\}$ be arbitrary maximal antichain in
$\mathds{P}$. Denote $P_{n}=\\{p_{k}\mid\,k\leq n\\}$. By (I) of (), for every
set $P_{n}$ there is some finite subset $R_{n}$ of $\mathds{P}$ such that
$R_{n}\perp P_{n}$ and $\forall p\in Q_{m}\,[p\perp P_{n}\Rightarrow\exists
r\in R_{n}\,(p\leq r)$. We can assume that $R_{n}\in[Q_{m}]^{<\omega}$ because
$(p\in Q_{m}$ and $p\leq r)\Rightarrow r\in Q_{m}$. Therefore

($\oplus$) $\forall n\in\omega\,\exists
R_{n}\in[Q_{m}]^{<\omega}\smallsetminus\,\\{\emptyset\\}\,[R_{n}\perp
P_{n}\text{ and }\forall p\in Q_{m}\,(p\perp P_{n}\Rightarrow\exists r\in
R_{n}\,(r\leq p))].$

By ($\oplus$ ‣ 3.1) for every $p\in R_{n+1}$ there is $r\in R_{n}$ such that
$r\leq p$ (because $p\in Q_{m}$ and $p\perp P_{n}$). In this way, by
induction, we prove that $\forall n\in\omega\,\forall p\in R_{n}\,\exists r\in
R_{0})\,(r\leq p)$. Denote by $R_{n}^{\prime}$ the set $\\{r\in
R_{0}\mid\,\exists p\in R_{n}\,(r\leq p)\\}$. Then ($\oplus$ ‣ 3.1) holds with
the sets $R_{n}^{\prime}$ instead of $R_{n}$ and therefore without loss of
generality, we can assume that $R_{n}\subseteq R_{0}$ for all $n\in\omega$.
Since $P$ is a maximal antichain, there is $m_{0}\in\omega$ such that $\forall
p\in R_{0}\,(p\not\perp P_{m_{0}})$. Then it follows that
$R_{m_{0}}=\emptyset$ because $R_{m_{0}}\perp P_{m_{0}}$ and
$R_{m_{0}}\subseteq R_{0}$. Then by ($\oplus$ ‣ 3.1) for $n=m_{0}$ we get
$\forall p\in Q_{m}\,(p\not\perp P_{m_{0}})$. This finishes the proof that
$Q_{m}$ is leaf-soft. ∎

######  Remark 3.12.

Consider the following variants of (I) of ():

  1. (I1)

$\forall m\in\omega\,\exists R\in[\mathds{P}]^{<\omega}\,\forall
p\in\mathds{P}\,[h(p)\leq m\Rightarrow\exists r\in R\,(p\leq r)]$.

  2. (I2)

$\forall m\in\omega\,\forall
q\in\mathds{P}\smallsetminus\\{1_{\mathds{P}}\\}\,\exists
R\in[\mathds{P}]^{<\omega}\,(R\perp q)$ and $\forall p\in\mathds{P}\,[(p\perp
q$ and $h(p)\leq m)\Rightarrow\exists r\in R\,(p\leq r)]$.

  3. (I3)

$\forall m\in\omega\,(\forall q^{\prime},q\in\mathds{P}$ with $q\leq
q^{\prime}$ and $q^{\prime}\not\leq^{*}q)\,\exists
R\in[\mathds{P}]^{<\omega}\,(R\leq q^{\prime}\textrm{\ and\ }R\perp q$) and
$\forall p\in\mathds{P}\,[(p\leq q^{\prime}$ and $p\perp q$ and $h(p)\leq
m)\Rightarrow\exists r\in R\,(p\leq r)]$.

Note that (I1) and (I2) are instances of (I) for $P=\emptyset$ and $|P|=1$,
respectively. One can observe that

  * •

$\text{(I)}\Rightarrow$ (I2);

  * •

(I3)${}\Leftrightarrow\forall q^{\prime}\in\mathds{P}$
$(\mathds{P}|q^{\prime}$ satisfies (I2));

  * •

if $\mathds{P}$ has the largest element, then (I3) $\Rightarrow$ (I2).

We also consider the following property of pairs $\langle\mathds{P},h\rangle$
where $\mathds{P}$ is a poset and and $h$ a height function on $\mathds{P}$:
Say that $\langle\mathds{P},h\rangle$ has _club property_ if:

($\clubsuit$) (I) if $\\{p_{n}\mid\,n<\omega\\}$ is decreasing and $\exists
m\in\omega\,\forall n\in\omega\,(h(p_{n})\leq m)$, then $\exists
p\in\mathds{P}\,\forall n\in\omega\,(p\leq p_{n})$; (II) $\forall
m\in\omega\,\forall
P\in[\mathds{P}]^{<\omega}\smallsetminus\\{\emptyset\\}\,\exists
R\in[\mathds{P}]^{<\omega}\,(R\perp P)$ and $\forall p\in\mathds{P}\,(h(p)\leq
m)\,[p\perp P\Rightarrow\exists r\in R\,(p\leq r)]$; (III) if
$p,q\in\mathds{P}$ are compatible, then there is $r\leq p,q$ so that $h(r)\leq
h(p)+h(q)$.

The proof of the coming lemma relies essentially on [BJ93, Lem. 1.2].

######  Lemma 3.13.

A ccc poset $\mathds{P}$ with a height function $h\colon\mathds{P}\to\omega$
satisfying  _( $\clubsuit$ ‣ 3.1)_ is $\sigma$-soft-linked.

###### Proof.

For $m\in\omega$ let

$Q_{m}:=\\{p\in\mathds{P}\mid\,h(p)\leq m\\}\textrm{\ and\
}Q:=\bigcup_{m\in\omega}Q_{m}.$

We verify (3) of 3.9 for $\theta=\omega$, for the height function $h$, and for
the function $g(n,m)=n+m$. The only nontrivial condition is (i), so we show
it, i.e., that each $Q_{m}$ is leaf-soft. To this, assume that
$\\{p_{n}\mid\,n\in\omega\\}\subseteq\mathds{P}$ is a maximal antichain in
$\mathds{P}$ and show that there is some $n\in\omega$ such that $\forall q\in
Q_{m}\,\exists j<n\,(q\,||\,p_{j})$. Suppose not. So for each $n\in\omega$ let
$R_{n}$ be a finite subset of $\mathds{P}$ obtained by employing (II) of
($\clubsuit$ ‣ 3.1) to the family $P=\\{p_{i}\mid\,i<n\\}$. For each
$n\in\omega$, enumerate $R_{n}$ as $\\{q_{j}^{n}\mid\,j<k_{n}\\}$ for
$k_{n}<\omega$.

By assumption, none of these sets can be empty and we can assume that
$q_{j}^{n}\in Q_{m}$ for all $j$, $n$. By applying (II) of ($\clubsuit$ ‣ 3.1)
they form an $\omega$-tree with finite levels concerning $``\leq"$. Then by
Konig’s lemma, there is a function $f\in\omega^{\omega}$ such that
$q^{0}_{f(0)}\leq q^{1}_{f(1)}\leq q^{3}_{f(3)}\leq\cdots.$ By (I) of
($\clubsuit$ ‣ 3.1) there is a condition $q\leq q_{f(n)}^{n}$ for all $n$,
contradicting the fact that $\\{p_{n}\mid\,n\in\omega\\}$ is a maximal
antichain. ∎

###  3.2. Weaknnes versions of Soft-linkedness

Continuing our study of softness, we collect now two new weak notions
concerning not adding predictors. The following definitions are related to 3.1
and point in this direction.

######  Definition 3.14.

Let $\mathds{P}$ be a forcing notion and $\theta$ be a cardinal number.

  1. (1)

We say that _$\mathds{P}$ is $\theta$-up-soft-linked_ if there is a sequence
$\langle Q_{\alpha}\mid\alpha<\theta\rangle$ of upwards closed leaf-linked
subsets of $\mathds{P}$ such that

    1. (a)

$\bigcup_{\alpha<\theta}Q_{\alpha}$ is dense in $\mathds{P}$, and

    2. (b)

3.1 (3) (3)(c) holds.

  2. (2)

We say that _$\mathds{P}$ is $\theta$-$\uparrow$up-soft-linked_ if there is an
increasing sequence $\langle Q_{\alpha}\mid\alpha<\theta\rangle$ of upwards
closed leaf-linked subsets of $\mathds{P}$ such that

    1. (a)

$\bigcup_{\alpha<\theta}Q_{\alpha}$ is dense in $\mathds{P}$ and

    2. (b)

3.1 (3) (3)(c) holds.

Notice that $\theta$-soft-linked $\Rightarrow\theta$-up-soft-linked
$\Rightarrow\theta$-$\uparrow$up-soft-linked.

Just as in 3.9, we have:

######  Lemma 3.15.

For a poset $\mathds{P}$ and cardinal number $\theta$ consider the following
properties:

  1. (1)

$\mathds{P}$ is $\theta$-up-soft-linked.

  2. (2)

There are a dense set $Q\subseteq\mathds{P}$ and functions $h:Q\to\theta$ and
$g:\theta\times\theta\to\theta$ with the following properties:

    1. (i)

For every $\alpha<\theta$ the set $\\{p\in Q\mid h(p)=\alpha\\}$ is leaf-
linked.

    2. (ii)

$\forall p,q\in Q$ $[p\parallel q\Rightarrow\exists r\in Q$ $(r\leq p,q$ and
$h(r)\leq g(h(p),h(q)))]$.

    3. (iii)

$\forall p,q\in Q$ $q\leq p\Rightarrow h(q)\geq h(p)$.

  3. (3)

There are a dense set $Q\subseteq\mathds{P}$ and functions $h:Q\to\theta$ and
$g:\theta\times\theta\to\theta$ with the following properties:

    1. (i)

For every $\alpha<\theta$, the set $\\{p\in Q\mid h(p)\leq\alpha\\}$ is leaf-
linked.

    2. (ii)

$\forall p,q\in Q$ $[p\parallel q\Rightarrow\exists r\in Q$ $(r\leq p,q$ and
$h(r)\leq g(h(p),h(q)))]$.

    3. (iii)

$\forall p,q\in Q\,(q\leq p\Rightarrow h(q)\geq h(p))$.

Then $\ref{up-softc}\Rightarrow\ref{up-softb}$ and $\ref{up-
softa}\Rightarrow\ref{up-softb}$; $\ref{up-softc}\Rightarrow\ref{up-softa}$
for regular $\theta$; and $\ref{up-softa}\Leftrightarrow\ref{up-
softb}\Leftrightarrow\ref{up-softc}$ for $\theta=\omega$.

###### Proof.

$\ref{up-softc}\Rightarrow\ref{up-softb}$ is trivial.

$\ref{up-softa}\Rightarrow\ref{up-softb}$ Let $\langle
Q_{\alpha}\mid\alpha<\theta\rangle$ be a sequence of upwards closed subsets of
$\mathds{P}$ witnessing that $\mathds{P}$ is $\theta$-up-soft-linked. The set
$Q=\bigcup_{\alpha<\theta}Q_{\alpha}$ is a dense subset of $\mathds{P}$. For
$p\in Q$ define

$h(p)=\min\\{\alpha<\theta\mid p\in Q_{\alpha}\\}.$

Condition (iii) holds because the sets $Q_{\alpha}$ are upwards closed.
Condition (i) holds because every set $\\{p\in Q\mid h(p)=\alpha\\}$ is a
subset of the leaf-linked set $Q_{\alpha}$ and is therefore leaf-linked. By
using (1) (1)(b) of 3.14 for every $\alpha,\beta<\theta$ define

$g(\alpha,\beta)=\min\\{\gamma<\theta\mid\,\forall p\in Q_{\alpha}\,\forall
q\in Q_{\beta}\,[p\parallel q\Rightarrow\exists r\in Q_{\gamma}\,(r\leq
p,q)]\\}.$

Assume $p,q\in Q$ and $p\parallel q$. Then $p\in Q_{h(p)}$ and $q\in Q_{h(q)}$
and so there is $r\in Q_{g(h(p),h(q))}$ such that $r\leq p$ and $r\leq q$. By
definition of $h$, $h(r)\leq g(h(p),h(q))$ and hence (ii) holds.

$\ref{up-softc}\Rightarrow\ref{up-softa}$ for regular $\theta$. By (i) and
(iii) for every $\alpha<\theta$ the set $Q_{\alpha}=\\{p\in
Q\mid\,h(p)\leq\alpha\\}$ upwards closed leaf-linked and
$Q=\bigcup_{\alpha<\theta}Q_{\alpha}$ is a dense subset of $\mathds{P}$. We
verify (1) (1)(b) of 3.14. Fix $\alpha,\alpha^{\prime}\leq\theta$. Define

$\alpha^{*}=\sup\\{g(\xi,\eta)\mid\xi,\eta\leq\max\\{\alpha,\alpha^{\prime}\\}\\}.$

Note that $\alpha^{*}<\theta$ because $\theta$ is regular. Assume $p\in
Q_{\alpha}$, $q\in Q_{\alpha^{\prime}}$, and $p\parallel q$. Then
$g(h(p),h(q))\leq\alpha^{*}$ because $h(p)\leq\alpha$ and
$h(q)\leq\alpha^{\prime}$. By (ii) there is $r\leq p,q$ such that $h(r)\leq
g(h(p),h(q))\leq\alpha^{*}$ and hence $r\in Q_{\alpha^{*}}$.

$\ref{up-softb}\Rightarrow\ref{up-softc}$ for $\theta=\omega$. Note that
$\\{p\in Q\mid\,h(p)\leq n\\}$ is a finite union of leaf-linked sets $\\{p\in
Q\mid\,h(p)=k\\}$ for $k\leq n$ and is therefore leaf-linked. ∎

The following lemma presents another result similar to the previous one.

######  Lemma 3.16.

Let $\mathds{P}$ be forcing notion and a cardinal number $\theta$. Assume that
$\mathds{P}$ is $\theta$-$\uparrow$up-soft-linked. Then there are a dense set
$Q\subseteq\mathds{P}$ and functions $h:Q\to\theta$ and
$g:\theta\times\theta\to\theta$ with the following properties:

  1. (i)

For every $\alpha<\theta$, the set $\\{p\in Q\mid h(p)\leq\alpha\\}$ is leaf-
linked.

  2. (ii)

$\forall p,q\in Q$ $[p\parallel q\Rightarrow\exists r\in Q$ $(r\leq p,q$ and
$h(r)\leq g(h(p),h(q)))]$.

  3. (iii)

$\forall p,q\in Q\,(q\leq p\Rightarrow h(q)\geq h(p))$.

If $\theta$ is regular, the converse holds.

###### Proof.

Let $\langle Q_{\alpha}\mid\alpha<\theta\rangle$ be an increasing sequence of
upwards closed subsets of $\mathds{P}$ witnessing that $\mathds{P}$ is
$\theta$-$\uparrow$up-soft-linked. The set
$Q=\bigcup_{\alpha<\theta}Q_{\alpha}$ is a dense subset of $\mathds{P}$. For
$p\in Q$ define

$h(p)=\min\\{\alpha<\theta\mid\,p\in Q_{\alpha}\\}.$

Condition (iii) holds because the sets $Q_{\alpha}$ are upwards closed.
Condition (i) holds because $\\{p\in Q\mid h(p)\leq\alpha\\}=Q_{\alpha}$. By
using (2) (2)(b) of 3.14 for every $\alpha,\beta<\theta$ define

$g(\alpha,\beta)=\min\\{\gamma<\theta\mid\,\forall p\in Q_{\alpha}\,\forall
q\in Q_{\beta}\,[p\parallel q\Rightarrow\exists r\in Q_{\gamma}\,(r\leq
p,q)]\\}.$

Assume $p,q\in Q$ and $p\parallel q$. Then $p\in Q_{h(p)}$ and $q\in Q_{h(q)}$
and so there is $r\in Q_{g(h(p),h(q))}$ such that $r\leq p$ and $r\leq q$. By
definition of $h$, $h(r)\leq g(h(p),h(q))$ and hence (ii) holds.

For the converse, let $Q_{\alpha}=\\{p\in Q\mid\,h(p)\leq\alpha\\}$ for
$\alpha<\theta$. By (i) and (iii), $\langle
Q_{\alpha}\mid\alpha<\theta\rangle$ is an increasing sequence of upwards
closed sets and $Q=\bigcup_{\alpha<\theta}Q_{\alpha}$ is a dense subset of
$\mathds{P}$. We now verify (2) (2)(b) of 3.14. Fix
$\alpha,\alpha^{\prime}\leq\theta$. Define

$\alpha^{*}=\sup\\{g(\xi,\eta)\mid\xi,\eta\leq\max\\{\alpha,\alpha^{\prime}\\}\\}.$

Notice first that $\alpha^{*}<\theta$ because $\theta$ is regular. Assume
$p\in Q_{\alpha}$, $q\in Q_{\alpha^{\prime}}$, and $p\parallel q$. Then
$g(h(p),h(q))\leq\alpha^{*}$ because $h(p)\leq\alpha$ and
$h(q)\leq\alpha^{\prime}$. By (ii) there is $r\leq p,q$ such that $h(r)\leq
g(h(p),h(q))\leq\alpha^{*}$ and hence $r\in Q_{\alpha^{*}}$. ∎

The above theorem in conjunction with 3.16, we immediately obtain:

######  Corollary 3.17.

Let $\theta$ be a regular cardinal. Then $\theta$-$\uparrow$up-soft-linked
matches $\theta$-up-soft-linked. ∎

##  4\. Not adding predictors

The purpose of this section is to deal with the preservation of $\mathfrak{e}$
along FS iterations of $\theta$-soft-linked posets, which is based on
Brendle’s ideas to prove the consistency of
$\mathfrak{e}<\mathfrak{e}_{\mathbf{ubd}}$ in [Bre95, Sec.4.3]. We shall
prove, in fact, that $\theta$-soft-linked posets do not add predictors, so
they do not increase $\mathfrak{e}$ (see Theorem 4.19). We also provide
various examples of $\sigma$-soft-linked posets in this section.

###  4.1. A brief revising of preservation theory

We begin with some basic definitions but necessary to develop our goodness
property:

We say that $\mathbf{R}=\langle X,Y,{\sqsubset}\rangle$ is a relational system
if it consists of two non-empty sets $X$ and $Y$ and a relation ${\sqsubset}$.

  1. (1)

A set $F\subseteq X$ is _$\mathbf{R}$ -bounded_ if $\exists\,y\in Y\
\forall\,x\in F\colon x\sqsubset y$.

  2. (2)

A set $D\subseteq Y$ is _$\mathbf{R}$ -dominating_ if $\forall\,x\in
X\,\exists\,y\in D\colon x\sqsubset y$.

  3. (3)

Let $M$ be a set. Say that $x$ is _$\mathbf{R}$ -unbounded over $M$_ if
$\forall\,y\in Y\cap M\colon\neg(x\sqsubset y)$.

We associate two cardinal characteristics with this relational system
$\mathbf{R}$:

$\displaystyle\mathfrak{b}(\mathbf{R})$
$\displaystyle:=\min\\{|F|\mid\,F\subseteq X\text{ is
}\mathbf{R}\text{-unbounded}\\}\text{ the \emph{unbounding number of
$\mathbf{R}$}, and}$ $\displaystyle\mathfrak{d}(\mathbf{R})$
$\displaystyle:=\min\\{|D|\mid\,D\subseteq Y\text{ is
}\mathbf{R}\text{-dominating}\\}\text{ the \emph{dominating number of
$\mathbf{R}$}.}$

We now introduce two important concepts before providing some examples of
$\sigma$-soft-linked posets. These concepts were originally introduced by
Judah and Shelah [JS90], they were improved by Brendle [Bre91], and
generalized by the author and Mejía [CM19, Sect. 4].

######  Definition 4.1.

We say that $\mathbf{R}=\langle X,Y,\sqsubset\rangle$ is a generalized Polish
relational system (gPrs) if

  1. (1)

$X$ is a Perfect Polish space,

  2. (2)

$Y=\bigcup_{e\in\Omega}Y_{e}$ where $\Omega$ is a non-empty set and, for some
Polish space $Z$, $Y_{e}$ is non-empty and analytic in $Z$ for all
$e\in\Omega$, and

  3. (3)

${\sqsubset}=\bigcup_{n<\omega}{\sqsubset_{n}}$ where
$\langle{\sqsubset_{n}}\mid n<\omega\rangle$ is some increasing sequence of
closed subsets of $X\times Y$ such that, for any $n<\omega$ and for any $y\in
Y$, $(\sqsubset_{n})^{y}=\\{x\in X\mid\,x\sqsubset_{n}y\\}$ is closed nowhere
dense.

If $|\Omega|=1$, we just say that $\mathbf{R}$ is a _Polish relational system
(Prs)_.

######  Remark 4.2.

By 4.1 (3), $\mathbf{C}_{\mathcal{M}(X)}\preceq_{\mathrm{T}}\mathbf{R}$.
Therefore, $\mathfrak{b}(\mathbf{R})\leq\mbox{\rm non}(\mathcal{M})$ and
$\mbox{\rm cov}(\mathcal{M})\leq\mathfrak{d}(\mathbf{R})$.

######  Definition 4.3.

A poset $\mathds{P}$ is $\theta$-$\mathbf{R}$-good if, for any
$\mathds{P}$-name $\dot{h}$ for a member of $Y$, there is a non-empty set
$H\subseteq Y$ (in the ground model) of size ${<}\theta$ such that, for any
$x\in X$, if $x$ is $\mathbf{R}$-unbounded over $H$ then $\Vdash
x\not\sqsubset\dot{h}$.

We say that $\mathds{P}$ is $\mathbf{R}$-good if it is
$\aleph_{1}$-$\mathbf{R}$-good. Notice that $\theta<\theta_{0}$ implies that
any $\theta$-$\mathbf{R}$-good poset is $\theta_{0}$-$\mathbf{R}$-good. Also,
if $\mathds{P}\lessdot\mathds{Q}$ and $\mathds{Q}$ is
$\theta$-$\mathbf{R}$-good, then $\mathds{P}$ is $\theta$-$\mathbf{R}$-good.

The preceding is a standard property associated with preserving
$\mathfrak{b}(\mathbf{R})$ small and $\mathfrak{d}(\mathbf{R})$ large after
forcing extensions.

Below, we now present some examples of good posets that will be used in the
proof of Theorem 4.25:

######  Example 4.4.

We indicate the type of posets that are good for the Prs of Cichoń’s diagram.

  1. (1)

The relational system
$\omega^{\omega}:=\langle\omega^{\omega},\omega^{\omega},\leq^{*}\rangle$ is
Polish. All of the forcing presented by Subsection 4.2 are examples of
$\omega^{\omega}$-good sets. More generally, $\sigma$-$\mathrm{Fr}$-linked
posets are $\omega^{\omega}$-good. (see [Mej19, BCM21]). In particular,
$\sigma$-soft-linked poset is $\omega^{\omega}$-good by Theorem 3.5.

  2. (2)

Define
$\Omega_{n}:=\\{a\in[2^{<\omega}]^{<\aleph_{0}}\mid\,\mathbf{Lb}(\bigcup_{s\in
a}[s])\leq 2^{-n}\\}$ (endowed with the discrete topology) and put
$\Omega:=\prod_{n<\omega}\Omega_{n}$ with the product topology, which is a
perfect Polish space. For every $x\in\Omega$ denote

$N_{x}:=\bigcap_{n<\omega}\bigcup_{m\geq n}\bigcup_{s\in x(m)}[s],$

which is clearly a Borel null set in $2^{\omega}$.

Define the Prs $\mathbf{Cn}:=\langle\Omega,2^{\omega},\sqsubset\rangle$ where
$x\sqsubset z$ iff $z\notin N_{x}$. Recall that any null set in $2^{\omega}$
is a subset of $N_{x}$ for some $x\in\Omega$, so $\mathbf{Cn}$ and
$\mathbf{C}_{\mathcal{N}}^{\perp}$ are Tukey-Galois equivalent. Hence,
$\mathfrak{b}(\mathbf{Cn})=\mbox{\rm cov}(\mathcal{N})$ and
$\mathfrak{d}(\mathbf{Cn})=\mbox{\rm non}(\mathcal{N})$.

Any $\mu$-centered poset is $\mu^{+}$-$\mathbf{Cn}$-good ([Bre91]). In
particular, $\sigma$-centered posets are $\mathbf{Cn}$-good.

  3. (3)

Let $\mathbf{M}:=\langle 2^{\omega},\mathds{I}\times 2^{\omega},\sqsubset^{\rm
m}\rangle$ where

$x\sqsubset^{\rm m}(I,y)\text{ iff }\forall^{\infty}n\colon
x\\!\\!\upharpoonright\\!\\!I_{n}\neq y\\!\\!\upharpoonright\\!\\!I_{n}.$

This is a Polish relational system and
$\mathbf{M}\cong_{\mathrm{T}}\mathbf{C}_{\mathcal{M}}$ (by [Bla10]).

Note that, whenever $M$ is a transitive model of $\mathrm{ZFC}$, $c\in
2^{\omega}$ is a Cohen real over $M$ iff $c$ is $\mathbf{M}$-unbounded over
$M$.

  4. (4)

Kamo and Osuga [KO14] define a gPrs with parameters
$\varrho,\rho\in\omega^{\omega}$, which we denote by
$\mathbf{Lc}^{*}(\varrho,\rho)$. The review of its properties is sufficient
for this paper’s purpose. Assume that $\varrho>0$ and $\rho\geq^{*}1$.

    1. (a)

$\mathbf{Lc}^{*}(\varrho,\rho)\preceq_{\mathrm{T}}\mathbf{Lc}(\varrho,\rho)$
[CM19, Lem. 4.21].

    2. (b)

Any $\theta$-centered poset is
$\theta^{+}$-$\mathbf{Lc}^{*}(\varrho,\rho)$-good [CM19, Lem. 4.24].

The Prs relational system defined below is inspired by [Bre95, Sec. 4.3].

######  Definition 4.5.

A pair $\pi=(\langle A_{k}\mid k\in\omega\rangle,\langle\pi_{k}\mid
k\in\omega\rangle)$ is called generalized predictor iff for all $k\in\omega$,

  * •

$A_{k}\subseteq[k,\omega)$ is finite,

  * •

$\pi_{k}$ is a function with $\mbox{\rm
dom}\pi_{k}=\\{\sigma\in\omega^{<\omega}\mid\,\mathrm{lh}(\sigma)\in
A_{k}\\}$, and

  * •

$\mbox{\rm ran}\pi_{k}\subseteq[\omega]^{<\omega}$.

Let $\Pi$ be the collection of generalized predictors. The original definition
of predicting is a particular instance of this notion (in the case
$A_{k}=\\{\ell_{k}\\}$, $\ell_{k}<\ell_{k+1}$ and $|\pi_{k}(\sigma)|=1$ for
$\sigma\in\omega^{\ell_{k}}$).

For $\pi\in\Pi$ and $f\in\omega^{\omega}$, we say

$\pi\textit{\ predicts\ }f~{}(\textrm{denoted by\
}f\in^{\textbf{pr}}\pi)\textrm{\ iff\
}\forall^{\infty}k\in\omega\,\exists\ell\in
A_{k}(f(\ell)\in\pi_{k}(f{\upharpoonright}\ell));$

otherwise, $f$  _evades_ $\pi$. Define the relational system
$\mathbf{E}=\langle\omega^{\omega},\Pi,\in^{\textbf{pr}}\rangle$.

###  4.2. Some examples of $\sigma$-soft-linked posets

Now we look at instances of $\sigma$-soft-linked posets. To illustrate the
next example, it is necessary to present the definition of  _$(b,h)$ -ED
(eventually different real) forcing_, denoted as $\mathds{E}^{h}_{b}$, which
was introduced by Kamo and Osuga in [KO14] and that it has been extensively
investigated in [CM19, BCM21, BCM23]. This forcing is used to increase
$\mathfrak{b}^{\mathrm{aLc}}_{b,h}$ (the _anti-localization cardinal_ , see
e.g. [CM23] for a definition and more).

######  Definition 4.6.

Fix $b:\omega\to(\omega+1)\smallsetminus\\{0\\}$ and $h\in\omega^{\omega}$
such that $\lim_{i\to+\infty}\frac{h(i)}{b(i)}=0$ (when $b(i)=\omega$,
interpret $\frac{h(i)}{b(i)}$ as $0$). Define the _$(b,h)$ -ED (eventually
different real) forcing_ $\mathds{E}^{h}_{b}$ as the poset whose conditions
are of the form $p=(s,\varphi)$ such that, for some $m:=m_{p}<\omega$,

  1. (1)

$s\in\mathrm{seq}_{<\omega}(b)=\bigcup_{n<\omega}\prod_{k<n}b(k)$,
$\varphi\in\mathcal{S}(b,m\cdot h)$, and

  2. (2)

$m\cdot h(i)<b(i)$ for every $i\geq|s|$,

ordered by $(t,\psi)\leq(s,\varphi)$ iff $s\subseteq t$, $\forall
i<\omega(\varphi(i)\subseteq\psi(i))$ and $t(i)\notin\varphi(i)$ for all
$i\in|t|\smallsetminus|s|$. For $s\in\mathrm{seq}_{<\omega}(b)$ and $m<\omega$
put

$E^{h}_{b}(s,m):=\\{(t,\varphi)\in\mathds{E}^{h}_{b}\mid\,t=s\text{\ and
}m_{(t,\varphi)}\leq m\\}.$

Denote $\mathds{E}_{b}:=\mathds{E}^{1}_{b}$,
$\mathds{E}:=\mathds{E}_{\omega}$, $E_{b}(s,m):=E^{1}_{b}(s,m)$, and
$E(s,m):=E_{\omega}(s,m)$.

It is not hard to see that $\mathds{E}_{b}^{h}$ is $\sigma$-linked. Even more,
whenever $b\geq^{*}\omega$, $\mathds{E}_{b}^{h}$ is $\sigma$-centered.

When $h\geq^{*}1$, $\mathds{E}_{b}^{h}$ adds an eventually different
real111More generally, $\mathds{E}_{b}^{h}$ adds a real $e\in\prod b$ such
that $\forall^{\infty}_{i<\omega}(e(i)\notin\varphi(i))$ for any
$\varphi\in\mathcal{S}(b,h)$ in the ground model. in $\prod b$.

######  Example 4.7.

Let $b,h$ be as in above. Then $\mathds{E}^{h}_{b}$ is $\sigma$-soft-linked.
In particular, $\mathds{E}$ is $\sigma$-soft-linked.

In point of fact, it is clear that $\bigcup_{(s,m)}E^{h}_{b}(s,m)$ is dense in
$\mathds{E}^{h}_{b}$ and by [BCM21, Lem. 3.8], it is known that
$E^{h}_{b}(s,m)$ is Fr-linked, so by 3.6 it is leaf-linked. Therefore, (3)(a)
and (3)(b) of 3.1 (3) is met.

On the other hand, given $m,m^{\prime}\in\omega$, and
$s,s^{\prime}\in\mathrm{seq}_{<\omega}(b)$. Wlog assumes that $s^{\prime}$ is
longer than $s$. Next, let $s^{*}:=s^{\prime}$ and $m^{*}:=m+m^{\prime}$. It
can be proved that for any $(t,\psi)\in E^{h}_{b}(s,m)$ and
$(t^{\prime},\psi^{\prime})\in E^{h}_{b}(s^{\prime},m^{\prime})$ compatible
each other then there is $(t^{*},\psi^{*})\in E^{h}_{b}(s^{*},m^{*})$
extending $(t,\psi)$ and $(t^{\prime},\psi^{\prime})$. Hence, (3)(c) of 3.1
(3) holds.

We now introduce a forcing to increase $\mathfrak{b}^{\mathrm{Lc}}_{b,h}$,
which typically adds _a slalom_. This forcing was introduced by Brendle and
Mejía [BM14].

######  Definition 4.8.

For $b,h\in\omega^{\omega}$ such that $(\forall i<\omega)$ $(b(i)>0$ and
$h(i)\leq b(i))$, $\lim_{i\to\infty}h(i)=\infty$, and
$\lim_{i\to\infty}h(i)/b(i)=0$ define $\mathds{LOC}_{b,h}$ the localization
forcing by

$\mathds{LOC}_{b,h}:=\\{(p,n)\mid\,p\in\mathcal{S}(b,h),\,n<\omega\textrm{\
and\ }\exists m<\omega\,\forall i<\omega\,(|p(i)|\leq m)\\},$

ordered by $(p^{\prime},n^{\prime})\leq(p,n)$ iff $n\leq n^{\prime}$,
$p^{\prime}{\upharpoonright}n=p$, and $\forall i<\omega(p(i)\subseteq
p^{\prime}(i))$.

######  Example 4.9.

$\mathds{LOC}_{b,h}$ is $\sigma$-soft-linked.

Indeed, for
$s\in\mathcal{S}_{<\omega}(b,h)=\bigcup_{k\in\omega}\prod_{n<k}[b(n)]^{\leq
h(n)}$, and $m<\omega$, define

$L_{b,h}(s,m):=\\{(p,n)\in\mathds{LOC}_{b,h}\mid\,s\subseteq
p,\,n=|s|,\textrm{\ and\ }\forall i<\omega\,(|p(i)|\leq m)\\}.$

Notice that
$\bigcup_{(s,m)\in\mathcal{S}_{<\omega}(b,h)\times\omega}\mathds{LOC}_{b,h}(s,m)$
is dense in $\mathds{LOC}_{b,h}$ and also note that $L_{b,h}(s,m)$ is Fr-
linked by [Mej19, Rem. 3.31] (see also [Car23, Lem. 4.10]). Since
$L_{b,h}(s,m)$ is Fr-linked, by 3.6, it is leaf-linked. Hence, (3)(a) and
(3)(b) of 3.1 (3) hold.

Let $m,m^{\prime}\in\omega$, and $s,s^{\prime}\in\mathcal{S}_{<\omega}(b,h)$.
Wlog assumes that $s^{\prime}$ is longer than $s$. Put $s^{*}:=s^{\prime}$ and
$m^{*}:=m+m^{\prime}$. If $(p,n)\in L_{b,h}(s,m)$ and
$(p^{\prime},n^{\prime})\in L_{b,h}(s^{\prime},m^{\prime})$ with
$(p,\varphi)\,\|\,(p^{\prime},\varphi^{\prime})$, then there is a
$(p^{*},n^{*})\in L_{b,h}(s^{*},m^{*})$ such that
$(p^{*},n^{*})\leq(p,n),(p^{\prime},n^{\prime})$. This shows (3)(c) of 3.1
(3).

We now proceed to present the meager forcing, denoted by $\mathds{M}$ proposed
by Judah and Shelah [JS90]. It was demonstrated by them that $\mathds{M}$ does
not add dominating reals and increases $\mbox{\rm non}(\mathcal{M})$.

######  Definition 4.10.

For $\eta\subseteq 2^{<\omega}$ denote $n_{\eta}=\sup\\{|s|\mid\,s\in\eta\\}$.
Let $T$ be the set of the all $\eta\subseteq 2^{<\omega}$ such that

  1. (a)

$n_{\eta}\in\omega$;

  2. (b)

if $v\in\eta$ and $k<|v|$, then $v{\upharpoonright}k\in\eta$; and

  3. (c)

if $v\in\eta$ and $|v|<n_{\eta}$, then $v{}^{\smallfrown}0\in\eta$ or
$v{}^{\smallfrown}1\in\eta$.

For $\eta\in T$ let $[\eta]=\\{x\in
2^{\omega}\mid\,x{\upharpoonright}n_{\eta}\in\eta\\}\subseteq 2^{\omega}$ and
$\tilde{\eta}=\\{u\in\eta\mid\,|u|=n_{\eta}\\}$. Define the _meager forcing
$\mathds{M}$_ as the set

$\mathds{M}=\\{(\eta,H)\mid\,\eta\in T\text{\ and\ }H\subseteq[\eta]\text{\ is
finite}\\}$

ordered by $(\nu,K)\leq(\eta,H)$ iff $\tilde{\eta}=\nu\cap 2^{n_{\nu}}$ and
$H\subseteq K$. It is not hard to see that $\mathds{M}$ is $\sigma$-centered.
Moreover, $\mathds{M}$ adds a Cohen real.

Let

$\mathds{M}^{*}=\\{(\eta,H)\in\mathds{M}\mid\,\forall
v\in\tilde{\eta}\,(|H\cap[v]|=1)\\}.$

Notice that $\mathds{M}^{*}$ is dense in $\mathds{M}$.

We prove that $\mathds{M}^{*}$ is $\sigma$-soft-linked, so $\mathds{M}$ is
too. First note (3)(a) of 3.1 holds and (3)(b) of 3.1 holds by Theorem 4.13.

Our focus in what follows is on proving Theorem 4.13. The following linkedness
is stronger than Fr-linkedness.

######  Definition 4.11 ([GMS16]).

Let $\mathds{P}$ be a poset, and let $D\subseteq\mathcal{P}(\omega)$ be a non-
principal ultrafilter. Say that $Q\subseteq\mathds{P}$ is a _$D$
-$\lim$-linked_ if there is a function $\lim^{D}\colon
Q^{\omega}\to\mathds{P}$ and a $\mathds{P}$-name $\dot{D}^{\prime}$ of an
ultrafilter extending $D$ such that, for any $\bar{q}=\langle
q_{i}:\,i<\omega\rangle\in Q^{\omega}$,

${\lim}^{D}\,\bar{q}\Vdash\dot{W}(\bar{q})\in\dot{D}^{\prime}.$

A set $Q\subseteq\mathds{P}$ has _uf_ -$\lim$-linked if it is $D$-lim-linked
for any ultrafilter $D$.

For an infinite cardinal $\theta$, the poset $\mathds{P}$ is  _uniformly
$\mu$-$D$-$\lim$-linked_ if $\mathds{P}=\bigcup_{\alpha<\theta}Q_{\alpha}$
where each $Q_{\alpha}$ is $D$-$\lim$-linked and the $\mathds{P}$-name
$\dot{D}^{\prime}$ above mentioned only depends on $D$ (and not on
$Q_{\alpha}$, although we have different limits for each $Q_{\alpha}$. When
these $Q_{\alpha}$ are _uf_ -$\lim$-linked, we say that $\mathds{P}$ is
_uniformly $\mu$-uf-$\lim$-linked_

######  Remark 4.12.

For instance, random forcing is $\sigma$-uf-linked (see [Mej19, Lem. 3.29]),
but it may not be $\sigma$-uf-$\lim$-linked (cf. [BCM21, Rem. 3.10]). It is
clear that any uf-$\lim$-linked set $Q\subseteq\mathds{P}$ is uf-linked, which
implies $\mathrm{Fr}$-linked.

Our objective is to demonstrate that $\mathds{M}^{*}$ is uniformly
$\sigma$-uf-$\lim$-linked, witnessed by

$M_{\eta}:=M^{*}_{\eta}=\\{(\eta^{\prime},H)\in\mathds{M}^{*}\mid\,\eta^{\prime}=\eta\\}$

for $\eta\in T$. Let $D$ be an ultrafilter on $\omega$, and $\bar{p}=\langle
p_{n}\mid n\in\omega\rangle$ be a sequence in $M^{*}_{\eta}$ with
$p_{n}=(\eta,H_{n})$. Let $m_{n}:=|\eta\cap 2^{n_{\eta}}|$. Considering the
lexicographic order $\lhd$ of $2^{\omega}$, let $\\{x_{n,k}\mid\,k<m_{n}\\}$
be a $\lhd$-increasing enumeration of $H_{n}$. Next, find $a_{0}\in D$ and
$m_{0}<\omega$ such that $H_{n}=\\{x_{n,k}\mid\,k\in m_{0}\\}$ for all $n\in
a_{0}$. For each $k<m_{0}$, define $x_{k}=\lim_{n}^{D}x_{n,k}$ in $2^{\omega}$
by

$x_{k}(i)=s\textrm{\ iff\ }\\{n\in a_{0}\mid\,x_{n,k}(i)=s\\}\in D,$

which matches the topological $D$-limit. Then, the $D$-limit of $H_{n}$ can be
defined as $H:=\\{x_{k}\mid\,k<m_{0}\\}$ and $\lim^{D}\bar{p}:=(\eta,H)$. It
is clear that this limit is in $M^{*}_{\eta}$.

######  Theorem 4.13.

The poset $\mathds{M}^{*}$ is uniformly $\sigma$-uf-$\lim$-linked: For any
ultrafilter $D$ on $\omega$, there is a $\mathds{M}^{*}$-name of an
ultrafilter $\dot{D}^{\prime}$ on $\omega$ extending $D$ such that, for any
$\eta\in T$, and $\bar{p}\in M_{\eta}^{\omega}$,
$\lim^{D}\bar{p}\Vdash\dot{W}(\bar{p})\in\dot{D}^{\prime}$.

It suffices to show the following to establish the previous theorem:

######  Claim 4.14.

Assume $M<\omega$, $\\{\eta_{k}\mid\,k<M\\}\subseteq T$,
$\\{\bar{p}^{k}\mid\,k<M\\}$ such that each $\bar{p}^{k}=\langle p_{k,n}\mid
n<\omega\rangle$ is a sequence in $M_{\eta_{k}}$, $q_{k}$ is the $D$-limit of
$\bar{p}^{k}$ for each $k<M$, and $q\in\mathds{M}^{*}$ is stronger than every
$q_{k}$. If $a\in D$ then $q$ forces that
$a\cap\bigcap_{k<M}\dot{W}(\bar{p}^{k})\neq\emptyset$.

###### Proof.

Write $p_{k,n}=(s_{k},t_{k},H_{k,n})$, $q_{k}=(s_{k},t_{k},H_{k})$ where each
$H_{k}=\\{x^{k}_{j}\mid\,j<m_{0}^{k}\\}$ is the $D$-limit of
$H_{k,n}=\\{x^{k,n}_{j}\mid\,j<m_{0}^{k}\\}$ (increasing $\lhd$-enumeration)
with $m_{0}^{k}\leq m_{k}$, for all $n\in a$ (wlog). Assume that $(\eta,H)\leq
q\leq q_{k}$ in $\mathds{M}$ for all $k<M$. Then $\tilde{\eta}_{k}=\eta\cap
2^{n_{\eta}}$ and $H_{k}\subseteq H$, so

$a_{k}=\\{n\in\omega\mid\,\forall
j<m_{0}^{k}\,(x^{k,n}_{j}{\restriction}n_{\eta}=x_{j}^{k}{\restriction}n_{\eta})\\}$

is in $D$. Hence, $a\cap\bigcap_{k<M}b_{k}\neq\emptyset$, so choose $n\in
a\cap\bigcap_{k<M}b_{k}$, and put $r:=(\eta,H^{\prime})$ where
$H^{\prime}:=H\cup\bigcup_{k<M}H_{k,n}$. Then $H_{k,n}\subseteq[\eta]$ because
for every $j<m_{0}^{k}$,
$x^{k,n}_{j}{\restriction}n_{\eta}=x_{j}^{k}{\restriction}n_{\eta}\in\tilde{\eta}$,
$H^{\prime}\subseteq[\eta]$, so $r$ is a condition in $\mathds{M}$ and $r$ is
stronger than $q$ and $p_{n,k}$ for any $k<M$. Since $\mathds{M}^{*}$ is dense
in $\mathds{M}$, we can find $r^{\prime}\leq r$, which implies that
$r^{\prime}$ forces $n\in a\cap\bigcap_{k<M}\dot{W}(\bar{p}^{k})$. ∎

We have seen that $M_{\eta}$ are uf-lim-linked, so by 4.12 $M_{\eta}$ are Fr-
linked. Hence, by 3.6, $M_{\eta}$ are leaf-linked, so the condition (3)(a) of
3.1 (3). It remains to prove (3)(c) of 3.1 (3). To see this, let $(\eta,H)\in
M_{\eta}$ and $(\eta^{\prime},H^{\prime})\in M_{\eta^{\prime}}$. Wlog assumes
that $\eta^{\prime}$ is longer than $\eta$. So put $\eta^{*}=\eta^{\prime}$.
It is not difficult to see that, if $(\eta,H)\in M_{\eta}$ and
$(\eta^{\prime},H^{\prime})\in M_{\eta^{\prime}}$ with
$(\eta,H)\,\|\,(\eta^{\prime},H^{\prime})$, then there is a
$(\eta^{*},H^{*})\in M_{\eta^{*}}$ such that
$(\eta^{*},H^{*})\leq(\eta,H),(\eta^{\prime},H^{\prime})$.

To state the next example, we need the definition of the Dirichlet–Minkowski
forcing, due to Bukovský, Recław, and Repický [BRaR91, Sec. 7]. They used this
forcing notion to increase the uniformity of the collection of the Dirichlet
sets ($D$-sets) of $[0,1]$. We define Dirichlet–Minkowski forcing by

$\mathbb{D}\mathbb{M}:=\\{(s,F)\mid\,s\in\omega^{<\omega}\textrm{\ and\
}F\in[[0,1]]^{<\omega}\\}.$

Ordered by $(s,F)\leq(t,H)$ iff $t\subseteq t$, $H\subseteq F$, and $\forall
i\in\mbox{\rm dom}(s\smallsetminus t)\,\forall x\in H\colon\parallel s(i)\cdot
x\parallel<\frac{1}{i+1}$. The poset $\mathbb{D}\mathbb{M}$ is
$\sigma$-centered, since for $s\in\omega^{<\omega}$ the set

$M_{s}=\\{(t,A)\in\mathbb{D}\mathbb{M}\mid\,t=s\\}$

centered and $\bigcup_{s\in\omega^{<\omega}}M_{s}=\mathbb{D}\mathbb{M}$.

We claim that $\mathbb{D}\mathbb{M}$ is $\sigma$-soft-linked. In order to see
this, we shall verify that the sets

$M_{s,m}=\\{(t,F)\in\mathbb{D}\mathbb{M}\mid\,t=s\textrm{\ and\ }|F|\leq
m\\}.$

$s\in\omega^{<\omega}$ and $m\in\omega$, witness that $\mathbb{D}\mathbb{M}$
is $\sigma$-soft-linked. We leave to the reader to check the (3)(b) and (3)(c)
of 3.1 (3). We show that (3)(a) of 3.1 (3). Because of 3.6, it suffices to
prove that the sets $M_{s,m}$ are Fr-linked.

As in the previous example, we proceed to show a version of Fr-linked. We
shall demonstrate that the sets

$M_{s,m}=\\{(t,F)\in\mathbb{D}\mathbb{M}\mid\,t=s\textrm{\ and\ }|F|\leq
m\\}.$

$s\in\omega^{<\omega}$ and $m\in\omega$, witness that $\mathbb{D}\mathbb{M}$
is uniformly $\sigma$-uf-$\lim$-linked. For an ultrafilter $D$ on $\omega$,
and $\bar{p}=\langle p_{n}\mid n\in\omega\rangle\in P_{s,t,m}$, we show how to
define $\lim^{D}\bar{p}$. Let $p_{n}=(s,t,F_{n})\in M_{s,m}$. Considering the
lexicographic order $\lhd$ of $[0,1]$, and let $\\{x_{n,k}\mid\,k<m_{n}\\}$ be
a $\lhd$-increasing enumeration of $F_{n}$ where $m_{n}\leq m$. Next find an
unique $m_{*}\leq m$ such that $A:=\\{n\in\omega\mid\,m_{n}=m_{*}\\}\in D$.
For each $k<m_{*}$, define $x_{k}:=\lim_{n}^{D}x_{n,k}$ in $[0,1]$ where
$x_{k}$ is the unique member of $[0,1]$ such that $\\{n\in
A\mid\,x_{n,k}=x_{k}\\}\in D$ (this coincides with the topological $D$-limit).
Therefore, we can think of $F:=\\{x_{k}\mid\,k<m_{*}\\}$ as the $D$-limit of
$\langle F_{n}\mid n<\omega\rangle$, so we define $\lim^{D}\bar{p}:=(s,t,F)$.
Note that $\lim^{D}\bar{p}\in M_{s,m}$.

######  Theorem 4.15.

The poset $\mathbb{D}\mathbb{M}$ is uniformly $\sigma$-uf-$\lim$-linked: If
$D$ is an ultrafilter on $\omega$, then there is a $\mathbb{D}\mathbb{M}$-name
of an ultrafilter $\dot{D}^{\prime}$ on $\omega$ extending $D$ such that, for
any $s\in\omega^{<\omega}$, $m\in\omega$ and $\bar{p}\in M_{s,m}^{\omega}$,
$\lim^{D}\bar{p}\Vdash W(\bar{p})\in\dot{D}^{\prime}$.

Proof of the following is sufficient to prove the above theorem.

######  Claim 4.16.

Assume $M<\omega$,
$\\{(s_{k},m_{k})\mid\,k<M\\}\subseteq\omega^{<\omega}\times\omega$,
$\\{\bar{p}^{k}\mid\,k<M\\}$ such that each $\bar{p}^{k}=\langle p_{k,n}\mid
n<\omega\rangle$ is a sequence in $M_{s_{k},m_{k}}$, $q_{k}$ is the $D$-limit
of $\bar{p}^{k}$ for each $k<M$, and $q\in\mathbb{D}\mathbb{M}$ is stronger
than every $q_{k}$. Then, for any $a\in D$, there are $n<\omega$ and
$q^{\prime}\leq q$ stronger than $p_{k,n}$ for all $k<M$ (i.e. $q^{\prime}$
forces $a\cap\bigcap_{k<M}\dot{W}(\bar{p}^{k})\neq\emptyset$).

###### Proof.

Write $p_{k,n}=(s_{k},F_{k,n})$, $q_{k}=(s_{k},F_{k})$ where each
$F_{k}=\\{x^{k}_{j}\mid\,j<m_{*,k}\\}$ is the $D$-limit of
$F_{k,n}=\\{x^{k,n}_{j}\mid\,j<m_{*,k}\\}$ (increasing $\lhd$-enumeration)
with $m_{*,k}\leq m_{k}$. Assume that $q=(s,F)\leq q_{k}$ in
$\mathbb{D}\mathbb{M}$ for all $k<M$. Let

$U_{k}:=\left\\{\langle x_{j}\mid j<m_{*,k}\rangle\;\middle|\;\forall
j<m_{*,k}\,\forall i\in\mbox{\rm dom}(s\smallsetminus s_{k})\colon\parallel
s(i)\cdot x_{j}\parallel<\frac{1}{i+1}\right\\}$

which is an open neighborhood of $\langle x^{k}_{j}\mid j<m_{*,k}\rangle$ in
$[0,1]^{m_{*,k}}$. Then

$b_{k}:=\left\\{n<\omega\;\middle|\;\forall j<m_{*,k}\,\forall i\in\mbox{\rm
dom}(s\smallsetminus s_{k})\colon\parallel s(i)\cdot
x_{j}^{k,n}\parallel<\frac{1}{i+1}\right\\}\in D.$

Hence, $a\cap\bigcap_{k<M}b_{k}\neq\emptyset$, so choose $n\in
a\cap\bigcap_{k<M}b_{k}$ and put $q^{\prime}=(s,F^{\prime})$ where
$F^{\prime}:=F\cup\bigcup_{k<M}F_{k,n}$. This is a condition in
$\mathbb{D}\mathbb{M}$ because $|F^{\prime}|\leq|F|+\sum_{k<M}m_{*,k}$.
Furthermore, $q^{\prime}$ is stronger than $q$ and $p_{n,k}$ for any $k<M$. ∎

In order to establish the consistency of
$\mathfrak{e}<\mathfrak{e}_{\mathbf{ubd}}$, Brendle [Bre95] introduced the
following forcing notion:

######  Definition 4.17.

Given a $b\in\omega^{\omega}$ define the forcing $\mathbb{PR}_{b}$ as the
poset whose conditions are of the form $p=(A,\langle\pi_{n}\mid n\in
A\rangle,F)$ such that

  1. (i)

$A\in[\omega]^{<\aleph_{0}}$,

  2. (ii)

$\forall n\in A$, $\pi_{n}\colon\prod_{m<n}b(m)\to b(n)$, and

  3. (iii)

$F$ finite set of functions and $\forall f\in F\,\exists
k\leq\omega\,(f\in\prod_{n<k}b(n))$, i.e., $F$ is a finite subset of
$\mathrm{seq}_{<\omega}(b)\cup\mathrm{seq}(b)$.

The order is defined by $(A^{\prime},\pi^{\prime},F^{\prime})\leq(A,\pi,F)$
iff

  1. (a)

$A\subseteq A^{\prime}$, $\pi\subseteq\pi^{\prime}$,

  2. (b)

if $A^{\prime}\neq A$ and $A\neq\emptyset$, then $\max
A<\min(A^{\prime}\smallsetminus A)$,

  3. (c)

$\forall f\in F\,\exists g\in F^{\prime}\,(f\subseteq g)$, and

  4. (d)

$\forall f\in F\,\forall n\in(A^{\prime}\smallsetminus A)\cap\mbox{\rm
dom}(f)\,(\pi^{\prime}_{n}(f{\restriction}n)=f(n))$.

Furthermore, $\mathbb{PR}_{b}$ is $\sigma$-centered (and thus in particular
ccc). In particular, if $(A^{\prime},\pi^{\prime},F^{\prime})\leq(A,\pi,F)$,
$f_{0},f_{1}\in F$, $n\in\mbox{\rm dom}(f_{0})\cap\mbox{\rm
dom}(f_{1})\smallsetminus A$, $f_{0}{\restriction}n=f_{1}{\restriction}n$, and
$f_{0}(n)\neq f_{1}(n)$, then $n\notin A^{\prime}$. For any two conditions
$p_{i}=(A_{p_{i}},\pi_{p_{i}},F_{p_{i}})$, $i\in 2$, in $\mathbb{PR}_{b}$,

$\displaystyle p_{0}\parallel p_{1}\Leftrightarrow{}$
$\displaystyle(A_{p_{0}}\cup
A_{p_{1}},\pi_{p_{0}}\cup\pi_{p_{1}},F_{p_{0}}\cup F_{p_{1}})\text{ is
a~{}condition}$ ($\boxtimes$) and this condition is a lower bound of $p_{0}$
and $p_{1}$ in $\mathbb{PR}_{b}$ $\displaystyle\Leftrightarrow{}$
$\displaystyle A_{p_{0}}=A_{p_{1}}\text{ and }\pi_{p_{0}}=\pi_{p_{1}}\text{ or
}$ $\displaystyle\exists i\in 2\ A_{p_{i}}\subsetneq A_{p_{1-i}}\text{ and
}\pi_{p_{i}}\subsetneq\pi_{p_{1-i}}\text{ and }\max
A_{p_{i}}<\min(A_{p_{1-i}}\smallsetminus A_{p_{i}})$
$\displaystyle\phantom{\exists i\in 2}\text{ and }\forall f\in F_{p_{i}}\
\forall n\in(A_{p_{1-i}}\smallsetminus A_{p_{i}})\cap\mbox{\rm dom}(f)\
(\pi_{p_{1-i}})_{n}(f{\restriction}n)=f(n).$

Let $G$ be a $\mathbb{PR}_{b}$-generic filter over $V$. In $V[G]$, define

$\pi_{\mathrm{gen}}:=\bigcup\left\\{\langle\pi_{n}\mid n\in
A\rangle\;\middle|\;\exists A,F\colon(A,\langle\pi_{n}\mid n\in A\rangle,F)\in
G\right\\}.$

Then $\pi_{\mathrm{gen}}\in\Pi_{b}$ and for every $f\in\prod b\cap
V\,(f=^{\textbf{pr}}\pi_{\mathrm{gen}})$.

Our aim now is to show that any poset of the form $\mathbb{PR}_{b}$ is
$\sigma$-soft-linked.

######  Lemma 4.18.

For any $b\in\omega^{\omega}$, $\mathbb{PR}_{b}$ is $\sigma$-soft-linked.

###### Proof.

Let $h:\mathbb{PR}_{b}\to\omega$ be defined by
$h(p)=\max(A_{p}\cup\\{|F_{p}|\\})$, which is easily to see that is a height
function where $p=(A_{p},\pi_{p},F_{p})$. We then verify that the pair
$\langle\mathds{P},h\rangle$ satisfies the conditions of the club property
($\clubsuit$ ‣ 3.1):

(I): Assume that $P=\\{p_{n}\mid n<\omega\\}$ is a decreasing sequence of
conditions in $\mathbb{PR}_{b}$ such that $\\{h(p_{n})\mid n<\omega\\}$ is
eventually constant. Then the sequence
$\\{(A_{p_{n}},\pi_{p_{n}},|F_{p_{n}}|)\mid n<\omega\\}$ is eventually
constant and eventually equal to some $(A,\pi,m)$. Therefore one can easily
observe that there is $F\in[\omega^{\omega}]^{m}$ such that $(A,\pi,F)$ is a
lower bound of $P$. This proves (I).

(III): If $p,q\in\mathbb{PR}_{b}$ are compatible, then $r=(A_{p}\cup
A_{q},\pi_{p}\cup\pi_{q},F_{p}\cup F_{q})$ is a lower bound of $p$ and $q$ and
$h(r)\leq h(p)+h(q)$. Therefore (III) holds.

It remains to prove (II). To this end, for $m,k\in\omega$ denote
$Q_{m}=\\{p\in\mathbb{PR}_{b}\mid\,h(p)\leq m\\}$ and
$R_{m,k}=\\{p{\restriction}k\mid\,p\in Q_{m}\\}$ where
$p{\restriction}k=(A_{p}\cap
k,\pi_{p}{\restriction}k,\\{f{\restriction}k\mid\,f\in F_{p}\\})$ for
$p\in\mathbb{PR}_{b}$. Every set $R_{m,k}$ is finite and for every $p\in
Q_{m}$, $p\leq p{\restriction}k$ and $h(p{\restriction}k)\leq h(p)\leq m$.
Thus it is easy to see that (II) of ($\clubsuit$ ‣ 3.1) implies (II’) where

  1. (II’)

$\forall m\in\omega\,\forall
P\in[\mathbb{PR}_{b}]^{<\omega}\smallsetminus\\{\emptyset\\}\,\exists k\geq
m\,\forall p\in Q_{m}\,(p\perp P\Rightarrow(p{\restriction}k)\perp P)$.

Therefore to conclude the proof it is enough to check (II’). To see this, let
$P\in[\mathbb{PR}_{b}]^{<\omega}\smallsetminus\\{\emptyset\\}$. Find $k\geq m$
such that $\bigcup_{q\in P}A_{q}\subseteq k$. We show that $k$ is as required.
Let $p\in Q_{m}$ be such that $p\perp P$ and let $q\in P$ be arbitrary.

  1. (i)

If $\pi_{p}\cup\pi_{q}$ is not a function, then $(p{\restriction}k)\perp q$
because $\mbox{\rm dom}(\pi_{q})=A_{q}\subseteq k$.

Let us assume that $\pi_{p}\cup\pi_{q}$ is a function. Denote
$p^{\prime}=(A_{p}\cup A_{q},\pi_{p}\cup\pi_{q},F_{p}\cup F_{q})$ and
$p^{\prime\prime}=(A_{p}\cup
A_{q},\pi_{p}\cup\pi_{q},(F_{p}{\restriction}k)\cup F_{q})$. Then
$p^{\prime},p^{\prime\prime}\in\mathbb{PR}_{b}$, $p^{\prime}\leq
p^{\prime\prime}$, and by using ($\boxtimes$ ‣ 4.17) of 4.17,

$\displaystyle p\perp q\Leftrightarrow\text{$p^{\prime}$ is not a~{}lower
bound of $p$ and~{}$q$;}$ $\displaystyle(p{\restriction}k)\perp
q\Leftrightarrow\text{$p^{\prime\prime}$ is not a~{}lower bound of
$p{\restriction}k$ and~{}$q$.}$

We assume that $p\perp q$ and hence,

$p^{\prime}\nleq p$ or $p^{\prime}\nleq q$.

Note that the relations $(p^{\prime}\nleq p$ or $p^{\prime}\nleq q)$ and
$(p^{\prime\prime}\nleq p{\restriction}k$ or $p^{\prime\prime}\nleq q)$ cannot
be justified by the violation of (c) of 4.17.

  1. (ii)

Assume that $(p^{\prime}\nleq p$ or $p^{\prime}\nleq q)$ is justified by the
violation of $(\beta)$, i.e.,

$A_{p}\cup A_{q}\neq A_{p}$ and $A_{p}\neq\emptyset$ and $\max
A_{p}\geq\min((A_{p}\cup A_{q})\smallsetminus A_{p})$ or $A_{p}\cup A_{q}\neq
A_{q}$ and $A_{q}\neq\emptyset$ and $\max A_{q}\geq\min((A_{p}\cup
A_{q})\smallsetminus A_{q})$.

Since all mentioned sets are subsets of $k$, this property gives
$(p^{\prime\prime}\nleq p{\restriction}k$ or $p^{\prime\prime}\nleq q)$ by
violation of (c) of 4.17. Therefore $(p{\restriction}k)\perp q$.

  1. (iii)

Assume that $(p^{\prime}\nleq p$ or $p^{\prime}\nleq q)$ is justified by a
violation of (b) of 4.17, i.e., one of the following conditions holds:

$\displaystyle\exists f\in F_{p}\ \exists n\in((A_{p}\cup A_{q})\smallsetminus
A_{p})\cap\mbox{\rm dom}(f)\ (\pi_{q})_{n}(f{\restriction}n)\neq f(n)\text{
or}$ $\displaystyle\exists f\in F_{q}\ \exists n\in((A_{p}\cup
A_{q})\smallsetminus A_{q})\cap\mbox{\rm dom}(f)\
(\pi_{p})_{n}(f{\restriction}n)\neq f(n).$

Since all mentioned sets are subsets of $k$ by replacing $F_{p}$ with
$F_{p}{\restriction}k$ we get $(p^{\prime\prime}\nleq p{\restriction}k$ or
$p^{\prime\prime}\nleq q)$ by violation of (b) of 4.17. Therefore
$(p{\restriction}k)\perp q$ also in this last case. ∎

###  4.3. Forcing the evasion number small

We now prove the main theorem of this section. This theorem generalizes all of
the examples shown in Subsection 4.2.

######  Theorem 4.19.

Let $\mathds{P}$ be a ccc forcing notion. Assume that $\mathds{P}$ is
$\theta$-$\mathrm{soft}$-linked. Then $\mathds{P}$ is
$\theta^{+}$-$\mathbf{E}$-good.

###### Proof.

Since $\mathds{P}$ is $\theta$-$\mathrm{soft}$-linked, by employing 3.8, there
is a dense subset $Q=\bigcup_{\alpha<\theta}Q_{\alpha}\subseteq\mathds{P}$
with all sets $Q_{\alpha}$ soft-linked and there is $g:\theta\times
Q\to\theta$ fulfills (2).

We prove that for every $\mathds{P}$-name $\dot{\pi}$ such that
$\Vdash_{\mathds{P}}\dot{\pi}\in\Pi$ there is a non-empty set $H\subseteq\Pi$
(in the ground model) of size ${<}\theta$ such that, for any
$f\in\omega^{\omega}$, if $f$ is $\mathbf{E}$-unbounded over $H$ then
$\Vdash_{\mathds{P}}f\not\in^{\mathbf{pr}}\dot{\pi}$. Without loss of
generality, we can assume that
$\mathds{P}=Q=\bigcup_{\alpha<\theta}Q_{\alpha}$.

Suppose that $\Vdash_{\mathds{P}}\dot{\pi}\in\Pi$ and that $\dot{\pi}$ is of
the form $(\langle\dot{A}_{k}\mid k\in\omega\rangle,\langle\dot{\pi}_{k}\mid
k\in\omega\rangle)$. For every $k\in\omega$ fix a maximal antichain
$\\{p_{k}^{j}\mid j\in\omega\\}$ deciding $\dot{A}_{k}$. For every
$k,j\in\omega$ let $A_{k}^{j}\in[\omega\smallsetminus k]^{<\omega}$ be such
that $p_{k}^{j}\Vdash\dot{A}_{k}=A_{k}^{j}$ and let $\\{\ell_{k,j,i}\mid
i<a_{k,j}\\}$ be the increasing enumeration of $A_{k}^{j}$. Since $Q_{\alpha}$
is soft-linked, for every $k\in\omega$ we can find $n_{k}^{\alpha}\in\omega$
such that

$\forall p\in Q_{\alpha}\ \exists j<n_{k}^{\alpha}\ p\parallel p_{k}^{j}.$

Denote $\eta_{k}^{j,\alpha}=g(\alpha,p_{k}^{j})$; then by applying (2),

($\varotimes_{1}$) $\forall p\in Q_{\alpha}\ \exists j<n_{k}^{\alpha}\ \exists
q\in Q_{\eta_{k}^{j,\alpha}}\ q\leq p,p_{k}^{j}.$

For every $k,j\in\omega$ by induction on $i<a_{k,j}$ we define

$\displaystyle p_{k,\sigma}^{j,\tau}\in\mathds{P}\text{ and
}\eta_{k,\sigma}^{j,\tau,\alpha}<\theta$ for $\sigma\in\omega^{\ell_{k,j,i}}$
and $\tau\in\omega^{i+1}$, $\displaystyle
n_{k,\sigma}^{j,\tau,\alpha}\in\omega$ for $\sigma\in\omega^{\ell_{k,j,i}}$
and $\tau\in\omega^{i}$

(in fact, we will need $\eta_{k,\sigma}^{j,\tau,\alpha}$ and
$n_{k,\sigma}^{j,\tau,\alpha}$ only for $j<n_{k}^{\alpha}$ and only for
finitely many bounded sequences $\tau$; “how many $\tau$” depends on
$\alpha$).

#### Case $i=0$

For $\sigma\in\omega^{\ell_{k,j,0}}$ let $\\{p_{k,\sigma}^{j,\langle
n\rangle}\mid n\in\omega\\}$ be a maximal antichain of conditions below
$p_{k}^{j}$ deciding the value of $\dot{\pi}_{k}(\sigma)$. Since all
$Q_{\beta}$’s are soft-linked we can find
$n_{k,\sigma}^{j,\emptyset,\alpha}\in\omega$ such that

$\forall p\in Q_{\eta_{k}^{j,\alpha}}\ p\leq p_{k}^{j}\Rightarrow\exists
n<n_{k,\sigma}^{j,\emptyset,\alpha}\ p\parallel p_{k,\sigma}^{j,\langle
n\rangle}.$

Denote $\eta_{k,\sigma}^{j,\langle
n\rangle,\alpha}=g(\eta_{k}^{j,\alpha},p_{k,\sigma}^{j,\langle n\rangle})$;
then by utilizing (2),

($\varotimes_{2}$) $\forall p\in Q_{\eta_{k}^{j,\alpha}}\ p\leq
p_{k}^{j}\Rightarrow\exists n<n_{k,\sigma}^{j,\emptyset,\alpha}\ \exists q\in
Q_{\eta_{k,\sigma}^{j,\langle n\rangle,\alpha}}\ q\leq
p,p_{k,\sigma}^{j,\langle n\rangle}.$

#### Case $i+1$

For every $\sigma\in\omega^{\ell_{k,j,i+1}}$ and $\tau\in\omega^{i+1}$ let
$\\{p_{k,\sigma}^{j,\tau^{\frown}\langle n\rangle}\mid n\in\omega\\}$ be a
maximal antichain of conditions below
$p_{k,\sigma{\restriction}\ell_{k,j,i}}^{j,\tau}$ deciding the value of
$\dot{\pi}_{k}(\sigma)$. Since all $Q_{\beta}$’s are soft-linked we can find
$n_{k,\sigma}^{j,\tau,\alpha}\in\omega$ such that

$\forall p\in Q_{\eta_{k,\sigma{\restriction}\ell_{k,j,i}}^{j,\tau,\alpha}}\
p\leq p_{k,\sigma{\restriction}\ell_{k,j,i}}^{j,\tau}\Rightarrow\exists
n<n_{k,\sigma}^{j,\tau,\alpha}\ p\parallel
p_{k,\sigma}^{j,\tau^{\frown}\langle n\rangle}.$

Denote $\eta_{k,\sigma}^{j,\tau^{\frown}\langle
n\rangle,\alpha}=g(\eta_{k,\sigma{\restriction}\ell_{k,j,i}}^{j,\tau,\alpha},p_{k,\sigma}^{j,\tau^{\frown}\langle
n\rangle})$; then by using (2),

($\varotimes_{3}$) $\forall p\in
Q_{\eta_{k,\sigma{\restriction}\ell_{k,j,i}}^{j,\tau,\alpha}}\ p\leq
p_{k,\sigma{\restriction}\ell_{k,j,i}}^{j,\tau}\Rightarrow\exists
n<n_{k,\sigma}^{j,\tau,\alpha}\ \exists q\in
Q_{\eta_{k,\sigma}^{j,\tau^{\frown}\langle n\rangle,\alpha}}\ q\leq
p,p_{k,\sigma}^{j,\tau^{\frown}\langle n\rangle}.$

This finishes the definitions of all $p_{k,\sigma}^{j,\tau}$,
$\eta_{k,\sigma}^{j,\tau,\alpha}$, and $n_{k,\sigma}^{j,\tau,\alpha}$.

Hence for $k,j\in\omega$, $i<a_{k,j}$, $\sigma\in\omega^{\ell_{k,j,i}}$, and
$\tau\in\omega^{i+1}$ the condition $p_{k,\sigma}^{j,\tau}$ decides
$\dot{\pi}_{k}(\sigma)$; let $A_{k,\sigma}^{j,\tau}\in[\omega]^{<\omega}$ be
such that
$p_{k,\sigma}^{j,\tau}\Vdash\dot{\pi}_{k}(\sigma)=A_{k,\sigma}^{j,\tau}$.

Now, for every $\alpha<\theta$ we define $\pi^{\alpha}=(\langle
A^{\alpha}_{k}\mid k\in\omega\rangle,\langle\pi_{k}^{\alpha}\mid
k\in\omega\rangle)\in\Pi$ as follows: Let
$A_{k}^{\alpha}=\bigcup_{j<n_{k}^{\alpha}}A_{k}^{j}$ and let
$\pi_{k}^{\alpha}:\bigcup_{n\in
A_{k}^{\alpha}}\omega^{n}\to[\omega]^{<\omega}$ be defined by

$\displaystyle\pi_{k}^{\alpha}(\sigma)=\bigcup\\{A_{k,\sigma}^{j,\tau}\mid{}$
$\displaystyle\exists j<n_{k}^{\alpha}\ \exists i<a_{k,j}$
$\displaystyle\sigma\in\omega^{\ell_{k,j,i}}\text{ and
}\tau\in\omega^{i+1}\text{ and }\forall m\leq i\
\tau(m)<n_{k,\sigma{\restriction}\ell_{k,j,m}}^{j,\tau{\restriction}m,\alpha}\\}.$

Let $f\in\omega^{\omega}$ and assume that $f\not\in^{\mathbf{pr}}\pi^{\alpha}$
for all $\alpha<\theta$. So it suffices to prove that:

######  Claim 4.20.

$\Vdash_{\mathds{P}}f\not\in^{\mathbf{pr}}\dot{\pi}$.

Towards a contradiction assume that there are $p\in\mathds{P}$ and
$k_{0}\in\omega$ such that $p\Vdash``\forall k\geq k_{0}$
$\exists\ell\in\dot{A}_{k}$ $f(\ell)\in\dot{\pi}_{k}(f{\restriction}\ell)$”.
Take $\alpha<\theta$ such that $p\in Q_{\alpha}$. By the choice of $f$ there
is $k\geq k_{0}$ such that $\forall\ell\in A_{k}^{\alpha}$
$f(\ell)\notin\pi_{k}^{\alpha}(f{\restriction}\ell)$. By using
($\varotimes_{1}$), there is $j<n_{k}^{\alpha}$ and there is a condition
$q_{k}^{j,\alpha}\in Q_{\eta_{k}^{j,\alpha}}$ such that $q_{k}^{j,\alpha}\leq
p,p_{k}^{j}$. Then by using ($\varotimes_{2}$) there is
$n_{0}<n_{k,f{\restriction}\ell_{k,j,0}}^{j,\emptyset,\alpha}$ and
$q_{k,f{\restriction}\ell_{k,j,0}}^{j,\langle n_{0}\rangle,\alpha}\in
Q_{\eta_{k,f{\restriction}\ell_{k,j,0}}^{j,\langle n_{0}\rangle,\alpha}}$ such
that $q_{k,f{\restriction}\ell_{k,j,0}}^{j,\langle n_{0}\rangle,\alpha}\leq
q_{k}^{j,\alpha},p_{k,f{\restriction}\ell_{k,j,0}}^{j,\langle n_{0}\rangle}$.

Using ($\varotimes_{3}$) by induction on $i<a_{k,j}$ define
$\tau\in\omega^{a_{k,j}}$ and a decreasing sequence of conditions
$\\{q_{k,f{\restriction}{\ell_{k,j,i}}}^{j,\tau{\restriction}(i+1),\alpha}\mid
i<a_{k,j}\\}$ below $q_{k,f{\restriction}\ell_{k,j,0}}^{j,\langle
n_{0}\rangle,\alpha}$ such that $\tau(0)=n_{0}$,
$q_{k,f{\restriction}{\ell_{k,j,i}}}^{j,\tau{\restriction}(i+1),\alpha}\in
Q_{\eta_{k,f{\restriction}{\ell_{k,j,i}}}^{j,\tau{\restriction}(i+1),\alpha}}$,
and
$q_{k,f{\restriction}{\ell_{k,j,i}}}^{j,\tau{\restriction}(i+1),\alpha}\leq
p_{k,f{\restriction}{\ell_{k,j,i}}}^{j,\tau{\restriction}(i+1)}$. (Assume that
$\tau{\restriction}(i+1)$ is defined. By ($\varotimes_{1}$) we find
$\tau(i+1)\leq
n_{k,f{\restriction}\ell_{k,j,i+1}}^{j,\tau{\restriction}(i+1),\alpha}$ and
$q_{k,f{\restriction}\ell_{k,j,i+1}}^{j,\tau{\restriction}(i+2),\alpha}\in
Q_{\eta_{k,f{\restriction}\ell_{k,j,i+1}}^{j,\tau{\restriction}(i+2),\alpha}}$
such that
$q_{k,f{\restriction}\ell_{k,j,i+1}}^{j,\tau{\restriction}(i+2),\alpha}\leq
q_{k,f{\restriction}\ell_{k,j,i}}^{j,\tau{\restriction}(i+1),\alpha},p_{k,f{\restriction}\ell_{k,j,i+1}}^{j,\tau{\restriction}(i+2)}$.)

Let $m=a_{k,j}-1$ and $q=q_{k,f{\restriction}\ell_{k,j,m}}^{j,\tau,\alpha}$.
It follows that $\forall i<a_{k,j}$ $q\leq
p_{k,f{\restriction}\ell_{k,j,i}}^{j,\tau{\restriction}(i+1)}\leq p_{k}^{j}$
where
$p_{k,f{\restriction}\ell_{k,j,i}}^{j,\tau{\restriction}(i+1)}\Vdash\dot{\pi}_{k}(f{\restriction}\ell_{k,j,i})=A_{k,f{\restriction}\ell_{k,j,i}}^{j,\tau{\restriction}(i+1)}$
and $p_{k}^{j}\Vdash\dot{A}_{k}=A_{k}^{j}$. Clearly $A_{k}^{j}\subseteq
A_{k}^{\alpha}$ and for every $i<a_{k,j}$,
$A_{k,f{\restriction}\ell_{k,j,i}}^{j,\tau{\restriction}(i+1)}\subseteq\pi^{\alpha}(f{\restriction}\ell_{k,j,i})$.
Therefore we get $q\Vdash\forall\ell\in\dot{A}_{k}$
$f(l)\notin\pi_{k}^{\alpha}(f{\restriction}\ell)\supseteq\dot{\pi}_{k}(f{\restriction}\ell)$
which is a contradiction. ∎

By a similar argument as in the proof of Theorem 4.19 and by applying 3.16, we
yield:

######  Theorem 4.21.

Every ccc $\theta$-$\uparrow$up-soft-linked forcing $\mathds{P}$ is
$\theta^{+}$-$\mathbf{E}$-good.

To conclude this section, we will demonstrate Brendle’s consistency of
$\mathfrak{e}<\mathfrak{e}_{\mathbf{ubd}}$ by utilizing the results we have
obtained so far.

In order to prove it, we begin with some notation:

To fix some notation, for two posets $\mathds{P}$ and $\mathds{Q}$, we write
$\mathds{P}\mathrel{\ooalign{$\subset$\cr\hbox{$\cdot\mkern
3.0mu$}\cr}}\mathds{Q}$ when $\mathds{P}$ is a complete suborder of
$\mathds{Q}$, i.e. the inclusion map from $\mathds{P}$ into $\mathds{Q}$ is a
complete embedding.

######  Definition 4.22 (Direct limit).

We say that $\langle\mathds{P}_{i}:\,i\in S\rangle$ is a _directed system of
posets_ if $S$ is a directed preorder and, for any $j\in S$, $\mathds{P}_{j}$
is a poset and $\mathds{P}_{i}\mathrel{\ooalign{$\subset$\cr\hbox{$\cdot\mkern
3.0mu$}\cr}}\mathds{P}_{j}$ for all $i\leq_{S}j$.

For such a system, we define its _direct limit_ $\mbox{\rm limdir}_{i\in
S}\mathds{P}_{i}:=\bigcup_{i\in S}\mathds{P}_{i}$ ordered by

$q\leq p{\ \mbox{$\Leftrightarrow$}\ }\exists\,i\in S\colon
p,q\in\mathds{P}_{i}\text{ and }q\leq_{\mathds{P}_{i}}p.$

FS iterations add Cohen reals, which is sometimes considered as a limitation
of the method.

######  Corollary 4.23 ([CM22, Cor. 2.7]).

Let $\pi$ be a limit ordinal of uncountable cofinality and let
$\mathds{P}_{\pi}=\langle\mathds{P}_{\alpha},\dot{\mathds{Q}}_{\alpha}\mid\alpha<\pi\rangle$
be a FS iteration of non-trivial posets. If $\mathds{P}_{\pi}$ is $\mbox{\rm
cf}(\pi)$-cc then it forces $\pi\preceq_{\mathrm{T}}\mathbf{M}$. In
particular, $\mathds{P}_{\pi}$ forces $\mbox{\rm
non}(\mathcal{M})\leq\mbox{\rm cf}(\pi)\leq\mbox{\rm cov}(\mathcal{M})$.

Good posets are preserved along FS iterations as follows.

######  Theorem 4.24 ([BCM23, Sec. 4]).

Let $\langle\mathds{P}_{\xi},\dot{\mathds{Q}}_{\xi}\mid\xi<\pi\rangle$ be a FS
iteration such that, for $\xi<\pi$, $\mathds{P}_{\xi}$ forces that
$\dot{\mathds{Q}}_{\xi}$ is a non-trivial $\theta$-cc
$\theta$-$\mathbf{R}$-good poset. Let
$\\{\gamma_{\alpha}\mid\,\alpha<\delta\\}$ be an increasing enumeration of $0$
and all limit ordinals smaller than $\pi$ (note that
$\gamma_{\alpha}=\omega\alpha$), and for $\alpha<\delta$ let
$\dot{c}_{\alpha}$ be a $\mathds{P}_{\gamma_{\alpha+1}}$-name of a Cohen real
in $X$ over $V_{\gamma_{\alpha}}$.

Then $\mathds{P}_{\pi}$ is $\theta$-$\mathbf{R}$-good. Moreover, if
$\pi\geq\theta$ then
$\mathbf{C}_{[\pi]^{<\theta}}\preceq_{\mathrm{T}}\mathbf{R}$,
$\mathfrak{b}(\mathbf{R})\leq\theta$ and $|\pi|\leq\mathfrak{d}(\mathbf{R})$.

The following theorem illustrates the effect of iterating forcing
$\mathds{P}_{b}$ on Cichoń’s diagram.

######  Theorem 4.25.

Let $\pi$ be an ordinal of uncountable cofinality such that
$|\pi|^{\aleph_{0}}=|\pi|$. The FS iteration of $\mathds{P}_{b}$ of length
$\pi$ (i.e. the FS iteration
$\langle\mathds{P}_{\alpha},\dot{\mathds{Q}}_{\alpha}\mid\alpha<\pi\rangle$
where each $\dot{\mathds{Q}}_{\alpha}$ is a $\mathds{P}_{\alpha}$-name of
$\mathds{P}_{b}$) forces

  1. (1)

$\mathfrak{c}=|\pi|$;

  2. (2)

$\mathbf{C}_{\mathcal{N}\\!\mathcal{A}}\cong_{\mathrm{T}}\mathbf{C}_{\mathcal{N}}^{\perp}\cong_{\mathrm{T}}\omega^{\omega}\cong_{\mathrm{T}}\mathbf{E}\cong_{\mathrm{T}}\mathbf{C}_{[\mathbb{R}]^{<\aleph_{1}}}$;
and

  3. (3)

$\mathbf{C}_{\mathcal{M}}\cong_{\mathrm{T}}\mathbf{E}_{b}\cong_{\mathrm{T}}\mbox{\rm
cf}(\pi)$ for any $b\in((\omega+1)\smallsetminus 2)^{\omega}$.

In particular, it is forced the Figure 1.

$\aleph_{1}$$\mbox{\rm add}(\mathcal{N})$$\mbox{\rm
cov}(\mathcal{N})$$\mbox{\rm non}(\mathcal{N})$$\mbox{\rm
cof}(\mathcal{N})$$\mbox{\rm add}(\mathcal{M})$$\mbox{\rm
cov}(\mathcal{M})$$\mbox{\rm non}(\mathcal{M})$$\mbox{\rm
cof}(\mathcal{M})$$\mathfrak{b}$$\mathfrak{d}$$\mathfrak{c}$$\mbox{\rm
non}(\mathcal{N}\\!\mathcal{A})$$\mathfrak{e}$$\mathfrak{e}_{\mathbf{ubd}}$$\aleph_{1}$$\mbox{\rm
cf}(\pi)$$|\pi|$ Figure 1. Cichon’s diagram after adding $\pi$-many generics
reals with $\mathds{P}_{b}$, where $\pi$ has uncountable cofinality and
$|\pi|^{\aleph_{0}}=|\pi|$.

###### Proof.

  1. ($\boxplus$)

Let $\\{b_{\xi}\mid\,\xi<\pi\\}$ be an enumeration of all the nice
$\mathds{P}_{\alpha}$-name for all members of $((\omega+1)\smallsetminus
2)^{\omega}$.

We define the iteration at each $\alpha$ as follows:

$\dot{\mathds{Q}}_{\alpha}:=\begin{array}[]{ll}\mathbb{PR}_{b_{\alpha}}&\text{if
$\dot{b}_{\alpha}$ is a $\mathds{P}_{\alpha+1}$-name for a function in
$((\omega+1)\smallsetminus 2)^{\omega}$,}\\\ \end{array}$

$\mathds{P}_{\alpha+1}=\mathds{P}_{\alpha}\ast\dot{\mathds{Q}}_{\alpha}$, and
$\mathds{P}_{\alpha}=\mbox{\rm limdir}_{\xi<\alpha}\mathds{P}_{\xi}$ if
$\alpha$ limit. At first, note that $\mathds{P}_{\pi}$ satisfies ccc and
forces $\mathfrak{c}=|\pi|$. Next, in view of 4.23, $\mathds{P}_{\pi}$ forces
$\pi\preceq_{\mathrm{T}}\mathbf{C}_{\mathcal{M}}$. Since
$\mathbf{C}_{\mathcal{M}}\preceq_{\mathrm{T}}\mathbf{E}_{b}$ (by 2.6), it
suffices to prove that, in $V_{\pi}$, there are maps $\Psi_{-}:\prod b\to\pi$
and $\Psi_{+}\colon\pi\to\Pi_{b}$ such that, for any $x\in\omega^{\omega}$,
and for any $\alpha<\pi$, if $\Psi_{-}(x)\leq\alpha$, then
$x=^{\textbf{pr}}\Psi_{+}(\alpha)$. To this end, for each $\alpha<\pi$, denote
by $\pi^{\mathrm{gen}}_{\alpha}$ the predictor generic real over $V_{\alpha}$
added by $\dot{\mathds{Q}}_{\alpha}$.

By ccc, there is $\alpha_{b}<\pi$ such that $b\in V_{\alpha_{b}}$. Next, for
$x\in\prod b\cap V_{\pi}$, we can find $\alpha_{b}\leq\alpha_{x}<\pi$ such
that $x\in V_{\alpha_{x}}$, so put $\Psi_{-}(x)=\alpha_{x}$. On the other
hand, for $\alpha<\pi$, when $\alpha\geq\alpha_{b}$, by ($\boxplus$) ‣ 4.3
there is an $\xi_{\alpha}$ such that $b=b_{\xi_{\alpha}}$, so define
$\Psi_{+}(\alpha)=\pi^{\mathrm{gen}}_{\xi_{\alpha}}$; otherwise, $\Psi_{-}(x)$
can be anything. It is clear that $(\Psi_{-},\Psi_{+})$ is the required Tukey
connection.

Using the fact that $\mathds{P}_{\pi}$ is obtained by FS iteration
$\langle\mathds{P}_{\alpha},\dot{\mathds{Q}}_{\alpha}\mid\alpha<\pi\rangle$
and that all its iterands are $\omega^{\omega}$-good, $\mathbf{Cn}$-good and
$\mathbf{E}$-good (see (1) and (2) of 4.4 and Theorem 4.19, respectively), so
by employing Theorem 4.24, $\mathds{P}_{\pi}$ forces
$\mathbf{C}_{[\omega^{\omega}]^{<\aleph_{1}}}\preceq_{\mathrm{T}}\omega^{\omega}$,
$\mathbf{C}_{[\omega^{\omega}]^{<\aleph_{1}}}\preceq_{\mathrm{T}}\mathbf{Cn}$
and
$\mathbf{C}_{[\omega^{\omega}]^{<\aleph_{1}}}\preceq_{\mathrm{T}}\mathbf{E}$,
respectively. Lastly, the opposite Tukey connections can be proved easily.

It remains to see that $\mathds{P}$ forces
$\mathbf{C}_{\mathcal{N}\\!\mathcal{A}}\cong_{\mathrm{T}}\mathbf{C}_{[\mathbb{R}]^{<\aleph_{1}}}$.
We just prove the nontrivial Tukey connection, i.e.,
$\mathbf{C}_{[\mathbb{R}]^{<\aleph_{1}}}\preceq_{\mathrm{T}}\mathbf{C}_{\mathcal{N}\\!\mathcal{A}}$.
Since all of the iterands of the iteration are $\sigma$-centered, by using 4.4
(4), they are $\mathbf{Lc}^{*}(\varrho,\rho)$-good, so by applying Theorem
4.24, $\mathds{P}_{\pi}$ forces
$\mathbf{C}_{[\mathbb{R}]^{<\aleph_{1}}}\preceq_{\mathrm{T}}\mathbf{Lc}^{*}(\varrho,\rho)$.
Moreover, by using (a) of 4.4 (4), $\mathds{P}_{\pi}$ forces
$\mathbf{C}_{[\mathbb{R}]^{<\aleph_{1}}}\preceq_{\mathrm{T}}\mathbf{Lc}(\varrho,\rho)$.
Finally, by the fact that in ZFC proves that
$\mathbf{Lc}(\varrho,\mathrm{id}_{\omega})\preceq_{\mathrm{T}}\mathbf{C}_{\mathcal{N}\\!\mathcal{A}}$
(see [CM23, Thm. 8.3]), we obtain that $\mathds{P}_{\pi}$ forces
$\mathbf{C}_{[\mathbb{R}]^{<\aleph_{1}}}\preceq_{\mathrm{T}}\mathbf{C}_{\mathcal{N}\\!\mathcal{A}}$.
This finished the proof of the theorem. ∎

##  5\. Open problems

One of our original intentions to introduce the notion of soft-linkedness was
to force the constellation of Figure 2. At point is that we do not know if
restricted versions of the forcing discussed in Subsection 4.2 behave well
with our preservation theory for $\mathfrak{e}$. Taking into account the
latter, the following is of interest:

######  Problem 5.1.

Could we modify our soft-linkedness property so that restrictions of the
forcing dealt with in Subsection 4.2 do not add predictors (so they won’t
increase $\mathfrak{e}$)?

Concening previous question, recently, Yamazoe [Yam24] introduced the notion
of  _closed-ultrafilter-limit_ (see [Yam24, Def. 3.26]) which keeps the
evasion number small and also can be considered on restrictions of forcings to
prove the constellation of Figure 2. This notion is related to Fr-linked
notion. So we ask:

######  Problem 5.2.

Is there any relationship between the notion of closed-ultrafilter-limit and
our notion of soft-linkedness?

In case 5.1 has a positive answer may force the constellation of Figure 2.

$\aleph_{1}$$\mbox{\rm add}(\mathcal{N})$$\mbox{\rm
cov}(\mathcal{N})$$\mbox{\rm non}(\mathcal{N})$$\mbox{\rm
cof}(\mathcal{N})$$\mbox{\rm add}(\mathcal{M})$$\mbox{\rm
cov}(\mathcal{M})$$\mbox{\rm non}(\mathcal{M})$$\mbox{\rm
cof}(\mathcal{M})$$\mathfrak{b}$$\mathfrak{d}$$\mathfrak{c}$$\mathfrak{e}$$\mathfrak{e}_{\mathbf{ubd}}$$\lambda_{1}$$\lambda_{2}$$\lambda_{3}$$\lambda_{4}$$\lambda_{5}$$\lambda_{6}$
Figure 2. Separation of the left side of Cichoń’s diagram with $\mathfrak{e}$.

######  Remark 5.3.

The constellation of Figure 2 was first forced by Goldstern, Kellner, Mejía,
and Shelah (unpublished). They used finitely additive measures (FAMs) along FS
iterations to construct a poset to force that

$\aleph_{1}<\mbox{\rm add}(\mathcal{N})<\mbox{\rm
cov}(\mathcal{N})<\mathfrak{b}<\mathfrak{e}<\mbox{\rm
non}(\mathcal{M})<\mbox{\rm cov}(\mathcal{M})<\mathfrak{d}<\mbox{\rm
non}(\mathcal{N})<\mbox{\rm cof}(\mathcal{N}).$

Though the previous work is unpublished, this was announced in [GKMS21].

On the other hand, consider random forcing $\mathds{B}$ as the poset whose
conditions are trees $T\subseteq 2^{<\omega}$ such that $\mathbf{Lb}([T])>0$
where $\mathbf{Lb}$ denotes the Lebesgue measure on $2^{\omega}$. The order is
$\subseteq$.

For $(s,m)\in 2^{<\omega}\times\omega$ set

$B(s,m):=\\{T\in\mathds{B}:[T]\subseteq[s]\text{\ and
}2^{|s|}\cdot\mathbf{Lb}([T])\geq 1-2^{-(1+m)}\\}.$

First of all, notice that $\bigcup_{(s,m)\in 2^{{<}\omega}\times\omega}B(s,m)$
is dense in $\mathds{B}$, and by [Mej19, Lem. 3.29] $B(s,m)$ is Fr-linked for
any $m\in\omega$ and $s\in 2^{<\omega}$, so (3)(a) and (3)(b) of 3.1 (3) hold.

Note that if $B(s,m)$’s work then also $B(k,m)$’s must work because $B(k,m)$
is the union of finitely many ($2^{k}$ many) of $B(s,m)$’s.

To see the condition (3)(c) of 3.1 (3), we want to show that given any $(k,m)$
and any $(k^{\prime},m^{\prime})$ we find $(k^{*},m^{*})$ in some way. Assume
that $T\in B(k,m)$ and $T^{\prime}\in B(k^{\prime},m^{\prime})$ are compatible
and let $s\in 2^{k}$, $s^{\prime}\in 2^{k^{\prime}}$ be such that
$[T]\subseteq[s]$ and $[T^{\prime}]\subseteq[s^{\prime}]$. Without loss of
generality assume that $k\leq k^{\prime}$ and hence $s\subseteq s^{\prime}$.
Then

$\mathbf{Lb}([T^{\prime}])\geq
2^{-k^{\prime}}\cdot(1-2^{-(m^{\prime}+1)})=2^{-k^{\prime}}-2^{-(m^{\prime}+k^{\prime}+1)}$

and

$\displaystyle\mathbf{Lb}([T]\cap[s^{\prime}])$
$\displaystyle\geq\mathbf{Lb}([T])-2^{-k^{\prime}}\cdot(2^{k^{\prime}-k}-1)$
$\displaystyle\geq
2^{-k}\cdot(1-2^{-(m+1)})-2^{-k^{\prime}}\cdot(2^{k^{\prime}-k}-1)$
$\displaystyle=2^{-k^{\prime}}-2^{-(m+k+1)}.$

It follows that

$\mathbf{Lb}([T\cap T^{\prime}])\geq
2^{-k^{\prime}}-2^{-(m^{\prime}+k^{\prime}+1)}-2^{-(m+k+1)}.$

We do not know whether there is a better estimation but this estimation does
not suffice to conclude that there is a suitable $(k^{*},m^{*})$: It many
happens that $m+k+1\leq k^{\prime}$ and we get a negative value. Then it is
unclear the condition (3)(c) of 3.1 (3). So we ask:

######  Problem 5.4.

Is it possible to obtain another representation for random forcing that allows
$\mathds{B}$ to be $\sigma$-soft-linked?

### Acknowledgments

I would like to thank Miroslav Repický for all the constructive suggestions on
the main concept in this paper that helped me to improve the concepts and
results of the preliminary version of this work, and for helping me with the
corrections of this material. I also would like to thank Diego Mejía for many
fruitful discussions relating to the material here.

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