1 Notation

The notation and terminology used here is mostly standard and it follows [Reference Chapital, González and Todorcevic10]. Given a set X and a (possibly finite) cardinal $\kappa $ , $[X]^\kappa $ denotes the family of all subsets of X of cardinality $\kappa $ . The sets $[X]^{<\kappa }$ and $[X]^{\leqslant \kappa }$ have the expected meanings. The family of all non-empty finite sets of X is denoted by $\text {Fin}(X)$ . That is, $\text {Fin}(X)=[X]^{<\omega }\backslash \{\emptyset \}$ . $\mathscr {P}(X)$ denotes the power set of X. By $\text {Lim}$ we mean the set of limit ordinals strictly smaller than $\omega _1$ . Given a partial order $(\mathbb {X},<)$ and $X,Y\subseteq \mathbb {X}$ , we write $X<Y$ whenever $x<y$ for each $x\in X$ and $y\in Y$ . In the same way, if $\mathcal {X},\mathcal {Y}\subseteq \mathscr {P}(\mathbb {X})$ , then $\mathcal {X}<\mathcal {Y}$ if and only if $X<Y$ for each $X\in \mathcal {X}$ and $Y\in \mathcal {Y}.$ Given $A,B\subseteq \mathbb {X}$ , A is said to be an initial segment of B, denoted as $A\sqsubseteq B$ , if for any $a\in A$ and $b\in B$ , if $b<a$ , then $b\in A$ . In this same situation we also say that B end-extends  $A.$

For a well-ordered set X (typically finite), we denote by $ot(X)$ its order type. We identify X with the unique strictly increasing function $h:ot(X)\longrightarrow X$ . In this way, $X(\alpha )=h(\alpha )$ denotes the $\alpha $ element of X with respect to its increasing enumeration. Analogously, $X[A]=\{X(\alpha )\,:\,\alpha \in A\}$ for $A\subseteq ot(X)$ . A family $\mathcal {D}$ is called a $\Delta $ -system with root R if $|\mathcal {D}|\geqslant 2$ and $X\cap Y= R$ whenever $X,Y\in \mathcal {D}$ are different. If moreover, $R<X\backslash R$ for any and $X\backslash R<Y\backslash R$ or vise-versa for any two $X,Y\in \mathcal {D}$ , we call $\mathcal {D}$ a root-tail-tail $\Delta $ -system.

Recall that if $f,g:\omega \longrightarrow \omega $ are distinct, we can define the number $\Delta (f,g)$ as the minimum k for which $f(k)\not =g(k).$ By convenience, we put $\Delta (f,g):=\omega $ whenever $f=g$ .

Given sets $U_{0},\dots , U_{n-1}$ , we define $U_0\otimes U_1:=\{\{u\ v\}\,:\,u\in U_0\text { and }v\in U_1\}$ and more generally $\otimes _{i<n} U_i:=\{\{u(0),\dots ,u(n-1)\}\,:\,u\in \prod \limits _{i<n}U_i\}$ . For $n\in \omega $ , $i\in 2$ and a function $\tau :n\longrightarrow 2$ , we define $\tau ^i:n\longrightarrow 2$ as $\tau ^i(j)=|i-\tau (j)|$ . Given $n,m\in \omega $ , $\sigma _0:n\longrightarrow 2$ and $\sigma _1:m\longrightarrow 2$ , we define the concatenation of $\sigma _0$ and $\sigma _1$ as the function $\sigma _0\frown \sigma _1:n+m\longrightarrow 2$ given by

$$ \begin{align*}\sigma_0\frown \sigma_1(i)=\begin{cases}\sigma_0(i)&\text{ if }i<n\\ \sigma_1(i-n)&\text{ if }i\geqslant n. \end{cases} \end{align*} $$

Whenever we say that M is an elementary submodel of $H(\kappa )$ , we mean that $(M,\in )$ is an elementary submodel of $(H(\kappa ),\in )$ , and whenever we say $\kappa $ is a large enough cardinal, we mean that $H(\kappa )$ contains every object of interest and satisfies enough ZFC axioms to carry the arguments discussed in the respective proof. For a topological space $(X,\tau )$ and $U\subseteq X$ , we denote the closure of U as $\overline {U}$ . Given $n\geqslant 1$ , we inherit $[\mathbb {R}]^n$ with the Vietoris topology whose base consist of all sets of the form $\otimes _{i<n} U_i$ where $U_i$ is open in $\mathbb {R}$ for each $i<n$ , and $U_i\cap U_j=\emptyset $ for any two distinct $i,j<n$ . It is well-known and easy to prove that $[\mathbb {R}]^n$ is homeomorphic to $\{(a_0,\dots ,a_{n-1})\in \mathbb {R}^n\,:\,a_0<\dots <a_{n-1}\}$ . In particular, this space is second countable.

Given a forcing notion $\mathbb {P}$ , we denote by $\mathfrak {m}(\mathbb {P})$ its Martin’s number, i.e., the minimal cardinal $\kappa $ for which there are $\kappa $ -many dense sets over $\mathbb {P}$ which are not intersected by any filter. The cardinal $\mathfrak {m}$ denotes the minimum of all $\mathfrak {m}(\mathbb {P})$ where $\mathbb {P}$ ranges over all $ccc$ -forcing notions. For us, MA denotes the statement $\mathfrak {m}=\mathfrak {c}>\omega _1$ .

2 Some preliminaries on entangled sets

This small section is mainly devoted to fix some notation regarding entangled sets as well as review some easy but important results regarding these objects. For a more detailed treatment we refer the reader to [Reference Guzmán and Todorcevic13].

Let $1\leqslant n \in \omega $ and $a,b\in [\mathbb {R}]^n$ be disjoint sets. We define $\text {type}(a,b):n\longrightarrow 2$ as

$$ \begin{align*}\text{type}(a,b)(i)=\begin{cases}0&\text{ if }a(i)<b(i)\\ 1&\text{ if }a(i)>b(i). \end{cases}\end{align*} $$

In the context of entangled sets, we refer to a function $\tau :n\longrightarrow 2$ as a (configuration) type. We say that a and b have type $\tau $ provided that $\text {type}(a,b)=\tau $ . Note that $\text {type}(a,b)$ , in case it is well-defined, is always distinct from $\text {type}(b,a)$ . In fact, $\text {type}(a,b)=\text {type}(b,a)^1.$ In order to take into consideration this fact, we define for an arbitrary $\tau $ as above, $\Vert \tau \Vert =\{\tau ^0,\tau ^1\}$ . In this way, $\Vert \text {type}(a,b)\Vert =\Vert \tau \Vert $ if and only if either $\text {type}(a,b)=\tau $ or $\text {type}(a,b)=\tau ^1.$ The following lemma is well-known and easy to prove.

The definition of the $\text {type}$ function can be further extended. Assume that $U, V\subseteq \mathscr {P}([\mathbb {R}]^n)$ are non-empty and such that $\bigcup U\cap \bigcup V=\emptyset $ . If $\tau :n\longrightarrow 2$ is such that $\text {type}(a,b)=\tau $ for any two $a\in U$ and $b\in V$ , we define $\text {type}(U,V)$ as $\tau $ . Whenever we assert that $\text {type}(U,V)=\tau $ , we will implicitly assume that U and V satisfy the requirements mentioned above. Furthermore, if $a\in [\mathbb {R}]^n$ and $V\subseteq \mathscr {P}([\mathbb {R}]^n)$ , then $\text {type}(\{a\},V)$ is simply written as $\text {type}(a,V).$

The following proposition is well-known. It implicitly says that “being disjoint and having type $\tau $ ” defines an open graph over $[\mathbb {R}]^n.$ We will frequently make use of this property without any explicit mention.

Proposition 2.2. Assume that $a,b\in [\mathbb {R}]^n$ are disjoint. Then there are open sets $U,V\subseteq [\mathbb {R}]^n$ so that $\bigcup U\cap \bigcup V=\emptyset $ , $a\in U$ , $b\in V$ , and

$$ \begin{align*}\text{type}(a,b)=\text{type}(U,V).\end{align*} $$

The next two lemmas already appear in some form in [Reference Abraham, Rubin and Shelah1, Reference Avraham and Shelah2]. Both of them are heavily used in the “duplication” type of arguments presented in [Reference Abraham, Rubin and Shelah1]. The first one is just a consequence of the fact that both $\min $ and $\max $ are definable. The second one was independently proven by Todorčević.

The main result of [Reference Guzmán and Todorcevic13] was the consistency of MA+PID+“There is a $2$ -entangled set.” In order to achieve this result, Guzmán and Todorčević proved a preservation theorem for Neeman iterations (as defined by Neeman in [Reference Neeman22]) and $2$ -entangled sets. It is easy to see that their argument actually serves to prove a much more general preservation theorem for uncountable graphs without uncountable independent sets. In particular, such preservation theorem can be written in terms of n-entangled sets as follows.

Theorem 2.5 [Reference Guzmán and Todorcevic13]

Let $2\leqslant n \in \omega $ , $\theta $ be an inaccessible cardinal, $J, \mathcal {S},\mathcal {T},\mathbb {P}^{\mathcal {S},\mathcal {T}}_\in $ , and $\mathbb {P}$ be such that:

• • $\mathcal {S}\subseteq \{M\in [H(\theta )]^\omega \,:\,M\text { is elementary submodel of }H(\theta )\}$ is stationary in $[H(\theta )]^\omega $ .

• • $\mathcal {T}=\{H(\lambda )\,:\,H(\lambda )\text { is elementary submodel of }H(\theta )\text { and }cof(\lambda )>\omega \}.$

• • $\mathbb {P}^{\mathcal {S},\mathcal {T}}_\varepsilon $ is the two type side condition $\varepsilon $ -collapse associated with $\mathcal {S}$ and $\mathcal {T}$ (see Definition 58 of [Reference Guzmán and Todorcevic13]).

• • $J:\theta \longrightarrow H(\theta )$ .

• • $\mathbb {P}=\mathbb {P}(J)$ is the Neeman iteration associated with J and $\mathbb {P}^{\mathcal {S},\mathcal {T}}_\varepsilon $ (see Definition 72 of [Reference Guzmán and Todorcevic13]).

Let $\mathbb {E}$ be an n-entangled set. If for every $X\in \mathcal {T}$ either X is trivial or

$$ \begin{align*}\mathbb{P}\cap X\Vdash"J(X)\text{ preserves the } n-\text{entangledness of }\mathbb{E},\!\mathrm{"}^3\end{align*} $$

Footnote then $\mathbb {P}\Vdash "{\mathbb {E}\text { is } n \text {-entangled}}.$ ”

As a direct consequence of this, we get a parametrized version of PFA for a fixed n-entangled set. Let us first explicitly describe this new axiom.

${\textbf {PFA}}_{\textbf {n-ent}}(\mathbb {E})$ : $\mathbb {E}$ is an n-entangled set and for any proper forcing $\mathbb {P}$ , if $\mathbb {P}$ preserves the n-entangledness of $\mathbb {E}$ , then $\mathfrak {m}(\mathbb {P})>\omega _1$ .

The same argument proved in Theorem 97 of [Reference Guzmán and Todorcevic13] yields the consistency of PFA $_{n-ent}(\mathbb {E})$ . This, as well as a general proper forcing axiom associated with families of graphs without independent sets, will appear in an upcoming paper which is joint work with Guzmán and Todorčević. Instances of these parametrized forcing axioms have been studied independently by Erlebach (see [Reference Erlebach12]).

In Theorem 37 of [Reference Guzmán and Todorcevic13] it was implicitly proved that $\mathbb {P}_{2-ent}(\mathbb {E})$ implies MA. It turns out that this result holds for arbitrary n’s. Even though the proof exposed in [Reference Guzmán and Todorcevic13] can be modified to yield the next theorem, we present a different proof here since ours is just a few lines long.

Theorem 2.7. $\mathbb {P}_{n-ent}(\mathbb {E})$ implies $\mathfrak {m}>\omega _1$ .

Proof. Let $\mathbb {Q}$ be a $ccc$ forcing. Let $\mathbb {P}_0$ be $\sigma $ -closed forcing collapsing $\mathfrak {c}$ to $\omega _1$ . Note that $\mathbb {P}_0$ preserves the n-entangledness of $\mathbb {E}$ and the fact that $\mathbb {Q}$ is $ccc$ . In this way, $\mathbb {P}_0\times \mathbb {Q}$ is a proper forcing. If it turns out that $\mathbb {P}_0\times \mathbb {Q}$ preserves the n-entangledness of $\mathbb {E}$ , we are done. This is because $\mathfrak {m}(\mathbb {P}_0\times \mathbb {Q})\leqslant \mathfrak {m}(\mathbb {Q})$ . On the other hand, if $\mathbb {P}_0\times \mathbb {Q}$ does not preserve the n-entangledness of $\mathbb {E}$ , it is necessarily true that

$$ \begin{align*}\mathbb{P}_0\Vdash"\mathbb{Q}\text{ does not preserve the } n-\text{entangledness of } \mathbb{E}."\end{align*} $$

This is due to the fact that n-entangled sets are preserved by $\sigma $ -closed forcings. Now, we evoke claim 1 of Lemma 8.5 from [Reference Abraham, Rubin and Shelah1].

Claim 1. [Reference Abraham, Rubin and Shelah1]: Assume CH holds and let $\mathbb {Q}$ be a ccc-forcing which does not preserve the n-entangledness of $\mathbb {E}$ . Then there is a $ccc$ -forcing $\mathbb {P}_1$ which preserves the n-entangledness of $\mathbb {E}$ and which adds an uncountable antichain to $\mathbb {Q}.$

Since $\mathbb {P}_0$ forces CH, we can fix a $\mathbb {P}_0$ -name $\dot {\mathbb {P}}_1$ for a $ccc$ -forcing satisfying the conclusion of the claim above. In order to finish, just note that $\mathbb {P}_0\ast \dot {\mathbb {P}}_1$ is a proper forcing which preserves the n-entangledness of $\mathbb {E}$ and adds an uncountable chain to  $\mathbb {Q}$ . As PFA $_{n-ent}(\mathbb {E})$ holds and being an uncountable antichain is a property which can be coded by $\omega _1$ -many dense sets, we conclude that $\mathbb {Q}$ is not $ccc$ . However, this is a contradiction.

Theorem 47 of [Reference Guzmán and Todorcevic13] can also be generalized to n-entangled sets. We do not prove such generalization since the arguments are identical to the original ones.

3 Destruction of $n+1$ -entangledness

The main purpose of this section is to prove Theorem 0.2. In order to do this, it suffices to show that PFA $_{n-ent}(\mathbb {E})$ implies the non-existence of $n+1$ -entangled sets. Fix $2\leqslant n\in \omega $ , an n-entangled set $\mathbb {E}$ , an uncountable $\mathbb {X}\subseteq \mathbb {R}$ and $\kappa =(2^{\mathfrak {c}})^+$ . We define $Sm(\kappa )$ as the set of all countable elementary submodels of $H(\kappa )$ having $\mathbb {E}, \mathcal {B}$ , and $\mathbb {X}$ as elements. We say that ${\mathcal {M}\in [Sm(\kappa )]^{<\omega }}$ is a chain whenever it is totally ordered with respect to $\in $ . In this situation, we write the elements of $\mathcal {M}$ increasingly as $\mathcal {M}(0),\dots ,\mathcal {M}(|\mathcal {M}|-1)$ . By convention, $\mathcal {M}(|\mathcal {M}|)$ denotes the universe.

Our goal is to define a proper forcing $\mathbb {P}(\mathbb {X},\mathbb {E})$ which preserves the n-entangledness of $\mathbb {E}$ and which force $\mathbb {X}$ to not be $n+1$ -entangled. The key of our proof lies in fixing the two unique constant types $\overline {0}$ and $\overline {1}$ over $[\mathbb {X}]^n$ as the ones we want to avoid, and construct via finite approximations an uncountable set $\mathcal {X}\subseteq [\mathbb {X}]^n$ such that the type of their elements is not constant. In order for this approach to work, it will be necessary that the elements of each $x\in \mathcal {X}$ are separated by certain chains of elementary submodels.

In the definition below, the x’s will latter be interpreted as possible candidates for elements of the set $\mathcal {X}\subseteq [\mathbb {X}]^n$ which will be forced by $\mathbb {P}(\mathbb {X},\mathbb {E})$ to avoid the constant types.

Note that $\in _\star $ is a partial order over $Gp(\kappa )$ . Whenever we talk about chains, initial segments, etc., inside $Gp(\kappa )$ we will refer to this order. We are now ready to define the required forcing notion.

The proof of the theorem below can be thought as an easier version of the proof of the preservation of the n-entangledness of $\mathbb {E}$ which will be presented in Theorem 3.10. For that reason, we leave it to the reader.

As a direct corollary of the two results above, we get the following corollary.

The next definition, together with the two lemmas below, constitute the keys to proving the preservation of the n-entangledness of $\mathbb {E}$ via the forcing $\mathbb {P}(\mathbb {X},\mathbb {E})$ .

Definition 3.6. Let $\mathcal {M}\in [Sm(\kappa )]^{n+1}$ and $a\in [\mathbb {E}]^{\leqslant n}$ . Given $i\leqslant n$ we define

$$ \begin{align*}a^{i\mathcal{M} }:=a\cap (\mathcal{M}(i+1)\backslash \mathcal{M}(i)),\end{align*} $$

We say that:

• • a is of the first kind over $\mathcal {M}$ if $|a^{i\mathcal {M}}|=1$ for each $i<n$ .Footnote ⁴

• • a is of the second kind over $\mathcal {M}$ if it is not of the first kind.

One main obstacle for proving the preservation of the n-entangledness of $\mathbb {E}$ is that at some point we will want to duplicate points in the second coordinates of good pairs, as well as $\leqslant n$ -size subsets of $\mathbb {E}$ via Lemmas 2.3 and 2.4, respectively. Unfortunately, we can not apply these two lemmas at the “same time.” The main motivation for defining sets of the first and second kind is to address this problem. Naively speaking if a set is of the first kind, we will prioritize duplications via Lemma 2.3. On the other hand, if we are dealing with a set of the second kind, we will prioritize duplications via Lemma 2.4. As it can bee seen in the next two lemmas, the main job of the chains of submodels forming the first coordinate of good pairs is to leave enough room of error for our arguments to work.

Lemma 3.7. Let $(\mathcal {M},x)\in Gp(\kappa )$ , $a\in [\mathbb {E}]^{ n}$ be of the first kind over $\mathcal {M}$ , and $\tau :n\longrightarrow 2$ be an arbitrary type. If $\mathcal {D}\in \mathcal {M}(0)\cap \mathscr {P}([X]^{n+1}\times [\mathbb {E}]^n)$ is such that ${(x,a)\in \mathcal {D}}$ , then there is $(y,b)\in \mathcal {D}\cap \mathcal {M}(0)$ so that:

1. (1) $\text {type}(x,y)$ is not constant;

2. (2) $\Vert \text {type}(a,b)\Vert =\Vert \tau \Vert $ .

Proof. For the sake of simplicity we will assume that $a^{i\mathcal {M}}<a^{j\mathcal {M}}$ whenever $i<j\leqslant n$ . Note that in this situation $a^{i\mathcal {M}}$ coincides with the set whose only element is the ith element of a with respect to its increasing order, namely $a(i)$ . Furthermore, we will assume that $\tau (n-1)=0$ . We proceed to build by reverse recursion two families of open subsets of $\mathbb {R}\, \langle U_i\rangle _{i\leqslant n}$ and $\langle V_i\rangle _{i<n}$ such that the following properties are satisfied:

1. (I) $U_0<\dots <U_{n-1}<U_n;$

2. (II) $V_0<\dots <V_{n-1}$ ;

3. (III) $\text {type}(x[\{n-1,n\}], U_{n-1}\otimes U_n)=\langle 0,1\rangle $ . In other words, $x(n-1)>U_{n-1}$ and $x(n)<U_n$ .

4. (IV) $\Vert \text {type}(a, \,\otimes _{i<n} V_i)\Vert =\Vert \tau \Vert $ .

Step n (Base step): Consider the set $\mathcal {D}_n=\{y\in \mathbb {X}\,:\,x(n-1)<y\text { and }(x[n]\cup \{y\},a)\in \mathcal {D}\}$ . Note that $x(n)\in \mathcal {D}_n$ , and since $\mathcal {D}\in \mathcal {M}(0)\subseteq \mathcal {M}(n)$ , it also holds that $\mathcal {D}_n\in \mathcal {M}(n)$ . According to the definition of good pair, $x(n)\not \in \mathcal {M}(n)$ . In this way, we can find a point $y\in \mathcal {D}_n\cap \mathcal {M}(n)$ so that $x<y$ . We define $U_n$ to be an open set inside $\mathcal {M}(0)$ such that $y\in U_n$ and $x(n)<\overline {U_n}.$

Step $n-1$ (Second step): Consider the set $\mathcal {D}_{n-1}$ of all $y\in \mathbb {X}$ so that $x(n-2)<y<\overline U_n$ and for which there are $z\in \mathbb {E}$ and $u\in [U_n]^1$ satisfying the following properties:

• • $a(n-2)<z;$

• • $(x[n-1]\cup \{y\}\cup u,a[n-1]\cup \{z\})\in \mathcal {D}$ .

Note that $x(n-1)\in \mathcal {D}_{n-1}$ and $\mathcal {D}_{n-1}\in \mathcal {M}(n-1) $ . Since $x(n-1)\not \in \mathcal {M}(n-1)$ , we can find $y\in \mathcal {D}_{n-1}\cap \mathcal {M}(n-1)$ so that $y<x(n-1)$ . Let $z\in \mathbb {E}\cap \mathcal {M}(n-1)$ witness the fact that $y\in \mathcal {D}_{n-1}$ . Observe that $z\not =a(n-1)$ because $a(n-1)\not \in \mathcal {M}(n-1)$ . We now define $U_{n-1}$ and $V_{n-1}$ to be open subsets of $\mathbb {R}$ inside $\mathcal {M}(0)$ so that:

• • $y\in U_{n-1}$ and $x(n-2)<\overline {U_{n-1}}<x(n-1)$ ;

• • $z\in V_{n-1}$ and $a(n-2)<\overline {V_{n-1}}$ ;

• • $a(n-1)<\overline {V_{n-1}}$ or $\overline {V_{n-1}}<a(n-1)$ . In this way, $\text {type}(a(n-1),V_{n-1})$ is well-defined.

We have now completed the first two steps of the recursion. Before we continue, let us define $\varepsilon \in 2$ as $0$ if $a(n-1)<\overline {V_{n-1}}$ , and as $1$ otherwise.

Recursive step: Assume that $k+1\leqslant {n-1}$ and we have defined $V_{k+1},\dots , V_{n-1},$ $U_{k+1},\dots ,U_{n}\in \mathcal {B}$ in such way that the following recursive hypotheses are fulfilled:

1. (1) $a(k)<\overline { V_{k+1}}<\dots <\overline {V_{n-1}}$ ;

2. (2) $x(k)<\overline {U_{k+1}}<\dots <\overline {U_{n-1}}<\overline {U_n}$ ;

3. (3) $\text {type}(x[\{n-1,n\}], U_{n-1}\otimes U_n)=\langle 0,1\rangle $ ;

4. (4) $\text {type}\big (a[n\backslash (k+1)], \: \bigotimes \limits _{k+1\leqslant i<n}V_i\,\big )(j)=\tau ^\varepsilon ((k+1)+j)$ for each $j<n-(k+1);$

5. (5) there are $u\in \bigotimes \limits _{k< i\leqslant n} U_i$ and $v\in \bigotimes \limits _{k<i<n} V_i$ so that $(x[k]\cup u,a[k]\cup v)\in \mathcal {D}.$

Let us define $\mathcal {D}_k$ as the set of all $z\in \mathbb {E}$ so thatFootnote ⁵ $a(k-1)<z<\overline {V_{k+1}}$ and for which there are $u\in \bigotimes \limits _{k< i\leqslant n} U_i$ , $v\in \bigotimes \limits _{k<i<n} V_i$ , and $y\in \mathbb {X}$ satisfying the following properties:

• • $x(k-1)<y<\overline {U_{k+1}};$

• • $(x[k-1]\cup \{y\}\cup u,a[k-1]\cup \{z\}\cup v)\in \mathcal {D}.$

Observe that $a(k)\in \mathcal {D}_k$ and $\mathcal {D}_k\in \mathcal {M}(k)$ . Since $\{a(k)\}=a^{i\mathcal {M}}$ , we also have that $a(k)\not \in \mathcal {M}(k)$ . In this way, we can find a point $z\in \mathcal {D}_k\cap \mathcal {M}(k)$ so that $a(k)<z$ if $\tau ^\varepsilon (k)=0$ and $z<a(k)$ if $\tau ^\varepsilon (k)=1.$ Let $y\in \mathbb {X}\cap \mathcal {M}(k)$ be a witness of the fact that $z\in \mathcal {D}$ . We now define $U_k$ and $V_k$ to be two open sets inside $\mathcal {M}(0)$ so that:

• • $y\in U_k$ and $x(k-1)<\overline {U_k}<\overline {U_{k+1}};$

• • $z\in V_k$ and $z(k-1)<\overline {V_k}<\overline {V_{k+1}};$

• • $a(k)<\overline {V_k}$ if $a(k)<z$ and $a(k)>\overline {V_k}$ if $z>a(k).$

By defining $U_k$ and $V_k$ in this way, it is easy to see that the recursive hypotheses of k are fulfilled. This finishes the recursion.

Note that $U_0,\dots ,U_n,V_0,\dots ,V_{n-1}$ have been constructed in such way that the properties (I)–(IV) are satisfied. Furthermore, by construction we have that

$$ \begin{align*}\left(\Big(\bigotimes_{i<n+1} U_i\Big)\times \Big( \bigotimes_{i<n} V_i \Big)\right)\cap \mathcal{D}\not=\emptyset.\end{align*} $$

As both $\mathcal {D}$ and $\mathcal {B}$ are elements of $\mathcal {M}(0)$ , there is a pair $(y,b)$ in the intersection above which also belongs to $\mathcal {M}(0)$ . It is clear that such $(y,b)$ finishes the job.

A particular instance of the lemma above happens when $a=\sigma =\emptyset $ . In such case, we can rewrite it in the following manner.

The proof of Theorem 0.2 will conclude once we prove the following theorem.

Theorem 3.10. $\mathbb {P}(\mathbb {X},\mathbb {E})$ preserves the n-entangledness of $\mathbb {E}.$

Proof. Let $\dot {\mathcal {A}}$ be a name for an uncountable subset of $[\mathbb {E}]^n$ whose elements are pairwise disjoint and let $q_0\in \mathbb {P}(\mathbb {X},\mathbb {E})$ . Fix $\theta $ a large enough cardinal so that $\mathbb {P}(\mathbb {X},\mathbb {E}),\kappa \in H(\theta )$ . Now, let $\hat {M}$ be an elementary submodel of $H(\theta )$ having $\mathbb {P}(\mathbb {X},\mathbb {E})$ , $\dot {\mathcal {A}}$ and $q_0$ as elements. Put $M=\hat {M}\cap H(\kappa ).$ By applying Lemma 3.3, we can find $x\in [\mathbb {X}]^{n+1}$ , and $p\leqslant q$ so that $(M,x)\in p$ . Since $\dot {\mathcal {A}}$ is forced to be uncountable and its elements are forced to be pairwise disjoint, we may assume without any loss of generality that there is $a\in [\mathbb {E}]^n$ disjoint from M for which $p\Vdash "{ a\in \dot {A} }$ .”

Given $i<n_p$ , let us write $\mathcal {M}^p_i$ and $x^p_i$ simply as $\mathcal {M}_i$ and $x_i$ , respectively. Also, fix $r:=M\cap p$ . Note that r is a condition end-extended by p. Furthermore, ${\mathcal {M}_{n_r}(0)=M.}$ Our plan is to find a condition $q\in \mathbb {P}(\mathbb {X},\mathbb {E})\cap M$ and $b\in [\mathbb {E}]^n$ so that:

• • q end-extends r and its compatible with p;

• • $q\Vdash {" a\in \dot {A} }"$ ;

• • $\Vert \text {type}(a,b)\Vert =\Vert \tau \Vert .$

In order to do this, we first fix $n_{rp}:=n_p-n_r$ and for each $i<$ we letFootnote ⁶

$$ \begin{align*}a_i:=(\mathcal{M}_{n_r+i+1}(0)\backslash \mathcal{M}_{n_r+i}(0))\cap a.\end{align*} $$

For the sake of simplicity we will assume that $a_i<a_j$ whenever $i<j<n_{rp}.$ Now, let $P=\{i<n_{rp}\:\,a_i\not =\emptyset \}$ and fix two sequences $\langle O_i\rangle _{i<{n_{rp}}}$ and $\langle W_i\rangle _{i\in P}$ inside M such that the following properties hold for any two distinct $i,j<n_{rp}$ :

• • $O_i$ is an open subset of $[\mathbb {R}]^{n+1}$ and $W_i$ is an open subset of $[\mathbb {R}]^{|a_i|}$ (whenever $i\in P)$ ;

• • $x_{n_r+i}\in O_i$ and $a_i\in W_i$ (for $i\in P)$ ;

• • $\overline {W_i}<\overline {W_j}$ whenever $i<j\in P$ ;

• • $\bigcup \overline {O_i}\cap \bigcup \overline {O_j}=\emptyset $ (So in particular, $\overline {O_i}\cap \overline {O_j}=\emptyset $ );

• • $\text {type}(O_i,O_j)=\text {type}(x_i,x_j),$

We proceed to define by reverse recursion two more sequences $\langle U_i\rangle _{i<n}$ and $\langle V_i\rangle _{i\in P}$ inside M satisfying the following conditions for $i<n$ :

1. (I) $U_i\subseteq O_i$ , it is open in $[\mathbb {R}]^{n+1}$ and $x_{n_r+i}\cap \bigcup U_i=\emptyset ;$

2. (II) $V_i\subseteq W_i$ and it is open in $[\mathbb {R}]^{|a_i|}$ and $a\cap \bigcup V_i=\emptyset $ whenever $i\in P$ ;

3. (III) $\text {type}(x_{n_r+i},U_i)$ is not constant;

4. (IV) $\Vert \text {type}(a,\otimes _{i\in P} V_i)\Vert =\Vert \tau \Vert $ ;

5. (V) there are $q\in \mathbb {P}(\mathbb {X},\mathbb {E})\cap M$ and $b\in (\otimes _{j\in P}V_j)\cap M$ such that:

   • • q end-extends r;

   • • $n_q=n_p$ ;

   • • $x^q_{n_r+i}\in U_i$ for each $i<n_{rp}$ ;

   • • $q\Vdash {"{b\in \dot {A} }}$ .”

It should be clear that for any q and b as in (V), conditions (I) and (III) assure that q is compatible with p. Furthermore, (IV) implies that $\Vert \text {type}(a,b)\Vert =\Vert \tau \rVert $ . Thus, the proof will be over once we finalize the construction of afore mentioned sequences.

Before we start the construction, we name one last set of objects. Let $m_{n_{rp}+1}=|a|$ and given $i\leqslant n_{rp}$ , let $m_i\leqslant |a|$ be such that $a[m_i]=a\cap \mathcal {M}_{n_{rp+i}} $ . Also let ${\sigma _i=: |a_i| \longrightarrow 2}$ be the type defined as

$$ \begin{align*}\sigma_i(j)=\tau(m_i+j).\end{align*} $$

For each $i\leqslant n_{rp}$ let $p_i=r\cup \{(\mathcal {M}_{n_r+j},x_{n_r+j})\,:j<i\}$ and note that $p_i$ is a condition in $\mathbb {P}(\mathbb {X},\mathbb {E})$ which is end-extended by p and such that $p_i\in \mathcal {M}_{i+1}(0).$ Furthermore, $p_{n_{rp}}=p$ and $p_0=r.$ Finally, let $\mathcal {D}$ be the set of all pairs

$$ \begin{align*}(\langle y_i\rangle_{i<{n_{rp}}}, b)\subseteq\Big(\prod_{i<n_{rp}} O_i\Big)\times\Big(\bigotimes_{i\in P}W\Big)\end{align*} $$

for which there is $q\in \mathbb {P}(\mathbb {X},\mathbb {E})$ end-extending r so that $n_q=n_p$ , $x^q_{n_r+i}=y_i$ for each i, and $q\Vdash "{ a\in \dot {A} }$ .” Observe that $D\in \hat {M}$ and $D\in H(\theta )$ . Thus, $D\in M=H(\theta )\cap \hat {M}$ . Even more, $(\langle x_{{n_{r}+i}}\rangle _{i<{n_rp}},a)\in \mathcal {D}$ .

The construction begins: Assume $k\leqslant n$ and we have defined $\langle U_i\rangle _{k<i<n}$ and $\langle V_i\rangle _{k<i\in P}$ inside M satisfying properties (I)–(III). Furthermore, assume that the following recursive hypotheses hold:

1. (IV) _r For any $k<i\in P$ , $\lVert \text {type}(a_i, V_i)\lVert =\lVert \tau \rVert $ if $a=a_i$ , and $\text {type}(a_i,V_i)=\sigma _i$ if $a\not =a_i.$

2. (V) _r There are $u\in \prod _{k<i<n_{rp}} U_i$ and $v\in \bigotimes _{k<i\in P}V_i$ so that $(\langle x_i\rangle _{i\leqslant n_r+k}\frown u, a[m_{k+1}]\cup v)\in \mathcal {D}.$

If $k\in P$ and $a_k\not =a$ , we define $\mathcal {D}_k$ as the set of all $(y,z)\in O_k\times W_k$ for which there are $u\in \prod _{k<i<n_{rp}} U_i$ and $v\in \bigotimes _{k<i\in P}V_i$ so that

$$ \begin{align*}(\langle x_i\rangle_{i<n_r+k}\frown y\frown u,a[m_{k}]\cup z\cup v)\in \mathcal{D}.\end{align*} $$

It should be clear that $\mathcal {D}\in \mathcal {M}_{n_r+k}(0)$ . Furthermore, $(x_{n_r+k},a_k)\in \mathcal {D}$ . Thus, we can use Lemma 3.8 to find $(y,b)\in \mathcal {D}_k\cap \mathcal {M}_{n_r+k}(0)$ so that $\text {type}(x_{n_r+k},y)$ is not constant and $\text {type}(a_k,z)= \sigma _k$ . Now, we just enlarge y and z to open sets $U_k\subseteq O_k$ and $V_k\subseteq W_k$ inside M whose union is disjoint from $x_{n_r+k}$ and $a_k$ , respectively, and such that $\text {type}(x,y)=\text {type}(x,U_k)$ and $\text {type}(a,z)=\text {type}(a, V_k)$ . It is straightforward that the recursive hypotheses still hold for k. Now, if $k\in P$ and $a_k=a$ we argue in a similar way but using Lemma 3.7, and if $k\not \in P$ we do the same but by making use of Lemma 3.9. This finishes the construction.

We have constructed two sequences $\langle U_i\rangle _{i\leqslant n}$ and $\langle V_i \rangle _{i\in P}$ which clearly satisfy (I)–(III) and (V). The key to prove that they also satisfy (IV) is to note one of the two following cases occur:

1. (1) $P=\{i\}$ for some $i\leqslant n$ , situation in which $a_i=a$ and consequently $\sigma _i=\tau $ . Thus $\Vert \text {type}( a,V_i)\Vert =\Vert \text {type}(a,\otimes _{i\in P} V_i)\Vert =\Vert \tau \Vert $ .

2. (2) $|P|\geqslant 2$ , which means that $a_i\not =a$ for each $i\in P$ . Thus, $\text {type}(a_i,V_i)=\sigma _i$ for each $i\in P$ . As $a_{P(0)}<\dots <a_{P(|P|-1)}$ and $V_{P(0)}<\dots <V_{P(|P|-1)}$ , we have that $\text {type}(a, \otimes _{i\in P} V_i)=\sigma _{P(0)}\frown \dots \frown \sigma _{P(|P|-1)}=\tau .$

As a corollary of the previous theorem, we get the main theorem of the article.

4 Some preliminaries on construction schemes

We dedicate this section present some preliminaries on construction schemes. All material presented here is spread throughout [Reference Chapital8–Reference Chapital, González and Todorcevic10, Reference Todorcevic28]. The reader is referred to the author’s PhD thesis [Reference Chapital7] for a complete introduction to this topic.

A (scheme) type is a sequence $\tau =\langle m_k,n_{k+1},r_{k+1}\rangle _{k\in \omega }$ of triplets of natural numbers such that the following conditions hold for any $k\in \omega $ :

1. (a) $m_0=1;$

2. (b) $n_{k+1}\geqslant 2;$

3. (c) $m_k>r_{k+1}; $

4. (d) $m_{k+1}=r_{k+1}+(m_k-r_{k+1})n_{k+1}.$

A construction scheme Footnote ⁷ (of type $\tau $ ) is a family $\mathcal {F}\subseteq \text {Fin}(\omega _1)$ which: Is cofinal in $(\text {Fin}(\omega _1),\subseteq )$ , any member of $\mathcal {F}$ has cardinality $m_k$ for some $k\in \omega $ , and furthermore, if we put $\mathcal {F}_k:=\{ F\in \mathcal {F}\,:\,|F|=m_k\}$ , then the following two properties are satisfied for each $k\in \omega $ :

1. (i) $\forall F,E\in \mathcal {F}_k\big (\; E\cap F\sqsubseteq E,F\;\big )$ ;

2. (ii) $\forall F\in \mathcal {F}_{k+1}\;\exists F_0,\dots ,F_{n_{k+1}-1}\in \mathcal {F}_k$ such that

   $$ \begin{align*}F=\bigcup\limits_{i<n_{k+1}}F_i.\end{align*} $$

   Moreover, $\langle F_i\rangle _{i<n_{k+1}}$ forms a root–tail–tail $\Delta $ -system with root $R(F)$ such that

   $$ \begin{align*}|R(F)|=r_{k+1}\ \mathrm{and}\ R(F)<F_0\backslash R(F)<\dots < F_{n_{k+1}-1}\backslash R(F).\end{align*} $$

Note that for a given $F\in \mathcal {F}_{k+1}$ , each of the $F_i$ ’s mentioned above can be written as $F[r_{k+1}]\cup F[\,[a_i,a_i+1)\,]$ where $a_i=r_{k+1}+i\cdot (m_{k+1}-m_k)$ . In particular, this is saying that the family $\langle F_i\rangle _{i<n_{k+1}}$ is unique, so we call it the canonical decomposition of F.

Definition 4.1 (The $\rho $ -function)

Let $\mathcal {F}$ be a scheme. We define $\rho :\omega _1^2\longrightarrow \omega $ as

$$ \begin{align*}\rho(\alpha,\beta)=\min (k\in\omega\,:\,\exists F\in \mathcal{F}_k\,(\{\alpha,\beta\}\subseteq F )\,).\end{align*} $$

For each finite $A\subseteq \omega _1$ , we also define

$$ \begin{align*}\rho^A=\max(\rho[A^2])=\max(\rho(\alpha,\beta)\,:\,\alpha,\beta\in A).\end{align*} $$

It is not hard to see that $\rho ^F=n$ for each $F\in \mathcal {F}_n$ . The most important feature of the function $\rho $ is that it is an ordinal metric. This means that it satisfies the properties stated in the following lemma.

Given $\alpha \in \omega _1$ and $k\in \omega $ we can define the k-closure of $\alpha $ as $(\alpha )_k:=\{\xi \leqslant \alpha \,:\rho (\alpha ,\xi )\leqslant k\}$ and $(\alpha )_k^-:=(\alpha )_k\backslash \{\alpha \}$ . Note that property $(om_4)$ is saying that all the k-closures are finite. It follows from the definition of a scheme that for any $k\in \omega $ there is at least one $F\in \mathcal {F}_k$ such that $\alpha \in F$ . Even more, for any such F we have the equalities:

$$ \begin{align*}F\cap (\alpha+1)=(\alpha)_k\end{align*} $$

$$ \begin{align*}F\cap \alpha=(\alpha)^-_k.\end{align*} $$

Given $\alpha \in \omega _1$ , the k-cardinality function $\lVert \alpha \rVert _{\_}:\omega \longrightarrow \omega $ is defined as

$$ \begin{align*}\lVert \alpha\rVert_k=|(\alpha)^-_k|.\end{align*} $$

Definition 4.3 (The $\Delta $ -function)

Let $\mathcal {F}$ be a scheme. We define $\Delta :\omega _1^2\longrightarrow \omega +1$ which is defined as

$$ \begin{align*}\Delta(\alpha,\beta)=\Delta(\lVert \alpha\rVert_{\_},\lVert \beta\rVert_{\_}).\end{align*} $$

Definition 4.4 (The $\Xi $ -function)

Let $\mathcal {F}$ be a scheme. Given $\alpha \in \omega _1$ we define the function $\Xi _\alpha :\omega \longrightarrow \{-1\}\cup \omega $ as follows; If $1\leqslant k\in \omega $ , then there is $F\in \mathcal {F}_k$ so that $\alpha \in F$ . According to property (ii) in the definition of a construction scheme, we have that either $\alpha \in R(F)$ or there is a unique $i<n_k$ such that $\alpha \in F_i\backslash R(F)$ . We then define

$$ \begin{align*}\Xi_\alpha(k):=\begin{cases}-1&\text{ if }\alpha\in R(F)\\ i&\text{ if }\alpha\in F_i\backslash R(F).\end{cases}\end{align*} $$

It can be proved that this definition does not depend on the choice of $F.$ Now, if $k=0$ , we simply define $\Xi _\alpha (k)$ as $0$ .

The next lemma relates the functions $\rho $ , $\Xi $ , and $\Delta $ in a very nice way.

Note that another way of stating $(xi_a)$ and $(xi_d)$ is simply by saying that $\Delta (\alpha ,\beta )=\Delta (\Xi _\alpha ,\Xi _\beta )$ .

The full capturing axiom (FCA) is the statement “There is a fully capturing scheme for any type.” We define in an analogous way the fully $\Delta $ -capturing axiom (FCA $^\Delta $ ). In [Reference Kalajdzievski and Lopez15], it was proved that FCA holds in the Cohen model. We proved the following theorem in [Reference Chapital, González and Todorcevic10].

In [Reference Chapital8], we proved the analogous result for fully- $\Delta $ -capturing schemes but assuming only CH.

For a scheme $\mathcal {F}$ , we say that a forcing notion $\mathbb {P}$ 2-preserves (resp $2$ - $\Delta $ -preserves) $\mathcal {F}$ whenever $\mathbb {P}$ forces $\mathcal {F}$ to be $2$ -capturing (resp. $2$ - $\Delta $ -capturing). We define $\mathfrak {m}_{\mathcal {F}}$ as the minimum of all Martin’s numbers associated with $ccc$ -forcing notions which $2$ -preserve $\mathcal {F}$ . Analogously $\mathfrak {m}^\Delta _{\mathcal {F}}$ denotes the minimum of all Martin’s numbers associated with $ccc$ -forcing notions which $2$ - $\Delta $ -preserve $\mathcal {F}$ . MA $(\mathcal {F})$ (resp. MA $^\Delta (\mathcal {F})$ ) denote the statement $\mathfrak {m}_{\mathcal {F}}=\mathfrak {c}>\omega _1$ (resp. $\mathfrak {m}^2_{\mathcal {F}}=\mathfrak {c}>\omega _1$ ).

In [Reference Chapital, González and Todorcevic9], we introduced and studied thoroughly the cardinal $\mathfrak {m}_{\mathcal {F}}$ . There we proved the following theorem.

It is easy to see that the exact same arguments yield an analogous theorem for $2$ - $\Delta $ -capturing schemes. The main tools for showing that a forcing $\mathbb {P}$ is $ccc$ and $2$ -preserves (resp. $2$ - $\Delta $ -preserve a scheme) are implicit in the equivalences presented above in Propositions 4.8 and 4.9. The following lemma is proved in [Reference Chapital, González and Todorcevic9].

In a completely analogous way we can prove the following lemma. We omit the proof.

5 n-entangled sets and full $\Delta $ -capturing

The main goal of this section is to show that the existence of a fully $\Delta $ -capturing scheme (with a fastly growing type) yields the existence of n-entangled sets which are not $n+1$ -entangled. Rather than constructing such entangled sets over the reals, we will do it over $\omega ^\omega $ endowed with the order $f<_{lex} g$ if $f(n)<g(n)$ where $n=\Delta (f,g)$ . It is easily seen that this total order is separable, hence it can be embedded in $\mathbb {R}.$ Now, the entangled sets which we construct will be indexed by countable ordinals. For that reason, we will need the equivalence presented below which the author assume is folklore (see [Reference Chapital8] for a proof).

For the rest of this section, we fix $n\in \omega $ and a fully $\Delta $ -capturing scheme $\mathcal {F}$ with type $\langle m_k,n_{k+1},r_{k+1}\rangle _{k\in \omega }$ in such way that $n_{k+1}\geqslant 2^{m_k}$ for each $k\in \omega $ . For any such k, let us enumerate (possibly with repetitions) the set $[m_k\backslash r_{k+1}]^{\leqslant n}$ as $\langle C^k_i\rangle _{0< i<n_{k+1}}$ . Given $\alpha \in \omega _1$ , we define two functions:

1. (1) $e^n_\alpha :\omega \longrightarrow \mathbb {Z}$ given by

   $$ \begin{align*}e^n_\alpha(k)=\begin{cases}0&\text{ if }\Xi_\alpha(k)\leqslant 0\\ \Xi_\alpha(k)&\text{ if }\Xi_\alpha(k)>0\text{ and }\lVert \alpha\rVert_{k-1}\in C^{k-1}_{\Xi_\alpha(k)}\\ \Xi_\alpha(k)&\text{ if }\Xi_\alpha(k)>0,\,|C^{k-1}_{\Xi_\alpha(k)}|=n\text{ and }\Vert \alpha\Vert_{k-1}<\min(C^{k-1}_{\Xi_\alpha(k)})\\ - \Xi_\alpha(k)&\text{ in any other case}. \end{cases}\end{align*} $$

2. (2) $o^n_\alpha :\omega \longrightarrow \mathbb {Z}$ given by

   $$ \begin{align*}o^n_\alpha(k)=\begin{cases}0&\text{ if }\Xi_\alpha(k)\leqslant 0\\ \Xi_\alpha(k)&\text{ if }\Xi_\alpha(k)>0\text{ and }\lVert \alpha\rVert_{k-1}\in C^{k-1}_{\Xi_\alpha(k)}\\ - \Xi_\alpha(k)&\text{ if }\Xi_\alpha(k)>0\text{ and }\Vert \alpha\Vert_{k-1}\notin C^{k-1}_{\Xi_\alpha(k)}. \end{cases}\end{align*} $$

Since n is fixed, let us write for now $e^n_\alpha $ and $o^n_\alpha $ simply as $e_\alpha $ and $o_\alpha $ , respectively. Also let $\mathbb {E}_e=\mathbb {E}^n_e:=\{e_\alpha \,:\,\alpha \in \omega _1\}$ and $\mathbb {E}^n_e=\mathbb {E}^n_o:=\{o_\alpha \,:\,\alpha \in \omega _1\}$ . We proceed to analyze the degree of entangledness of these two sets.

Proposition 5.2. $\mathbb {E}_e$ is $2n$ entangled.

Proof. Let $\tau :2n\longrightarrow 2$ be a type and $\mathcal {A}\subseteq [\omega _1]^{2n}$ be an uncountable family with pairwise disjoint elements. By changing $\tau $ by $\tau ^1$ if necessary, we may assume that $\tau ^{-1}[\{0\}]\leqslant n$ . Furthermore, if $\tau ^{-1}[\{0\}]=n$ we may also assume that $\tau (0)=0$ . As  $\mathcal {F}$ is fully $\Delta $ -capturing, there are $l\in \omega $ and a family $\{A_0,\dots ,A_{n_l-1}\}\in [\mathcal {A}]^{n_l}$ which is $\Delta $ -captured at level l. Let

$$ \begin{align*}C:=\{\lVert A_0(i) \Vert_{l-1}\,:\,i<2n\text{ and }\tau(i)=0\}.\end{align*} $$

Then $C\subseteq m_{l-1}\backslash r_l$ and $|C|\leqslant n$ . In this way, there is $j<n_l$ so that $C=C^{l-1}_j$ . We claim $\Vert \bar {\text {type}}(A_0,A_j)\Vert =\Vert \tau \Vert $ . Indeed, let $i<2n$ . Then:

1. (1) $e_{A_0(i)}(l)=0$ because $\Xi _{A_0(i)}(l)=0$ ,

2. (2) $\lVert A_0(i)\Vert _m=\Vert A_j(i)\Vert _m$ for any $m<l$ because $\Delta (A_0(i),A_j(i))=l$ .

From the two points above it easily follows that $e_{A_0(i)}(m)=e_{A_j(i)}(m)$ for any such m. In order to finish, just observe that:

• • If $\tau (i)=0$ , then $j=\lVert A_j(i)\Vert _{l-1}=\Vert A_0(i)\Vert _{l-1}\in C=C^{l-1}_j$ . Thus, $e_{A_j(i)}(l)=j>0=e_{A_0(i)}(l)$ .

• • If $\tau (i)=1$ , then $j=\lVert A_j(i)\Vert _{l-1}=\Vert A_0(i)\Vert _{l-1}\notin C=C^{l-1}_j$ . In this way, $e_{A_j(i)}=-j<0=e_{A_0(i)}(l)$ ( If $|C|<n$ , this is clear by definition. If $|C|=n$ , this is because we assume that $\tau (0)=0$ ).

In a completely similar way, we can prove the following result.

We proceed to bound from above the maximum amount of entangledness of both $\mathbb {E}_e$ and $\mathbb {E}_o$ . What we will do is to take an uncountable family of pairwise disjoint intervals over $\omega _1$ and show that we can refine that family in order to avoid the alternating types (those for which any two consecutive numbers have distinct value).

Proposition 5.4. $\mathbb {E}_e$ is not $2n+1$ entangled.

Proof. For each $\gamma \in \text {Lim}$ , let $D_\gamma :=[\gamma ,\gamma +2n+1)=(\gamma +2n+1)\backslash \gamma $ . By the pigeonhole principle, we can find $X\in [\omega _1]^{\omega _1}$ and $k,a\in \omega $ so that $\rho ^{D_\gamma }=k$ for each $\gamma \in X$ and $\lVert D_\gamma (0)\Vert _k=\Vert \gamma \Vert _k=a$ . We claim that for any two distinct $\delta ,\gamma \in X$ we have that $\Vert \bar {\text {type}}(D_\delta ,D_\gamma )\Vert \not =\tau $ where $\tau :2n+1\longrightarrow 2$ is the alternating type defined by

$$ \begin{align*}\tau(i)=\begin{cases} 0&\text{ if } i \text{ is even}\\ 1&\text{ if } i \text{ is odd}. \end{cases}\end{align*} $$

Assume towards a contradiction that this is not true. That is, there are $\gamma ,\delta \in X$ with $\Vert \bar {\text {type}}(D_\delta ,D_\gamma )\Vert =\tau $ . Let $l=\Delta (\gamma ,\delta )$ and note that since $\Vert \gamma \Vert _k=\Vert \delta \Vert _k$ , then $l>k$ .

Claim 1. For any $i<2n+1$ , $\Delta (\gamma +i,\delta +i)=l$ .

Proof of claim

Note that for any $k<s\in \omega $ , then there is $F\in \mathcal {F}_s$ with $[\gamma ,\gamma +2n+1)\subseteq F$ . By definition, we have that

$$ \begin{align*}\lVert \gamma+i\Vert_s=|(\gamma+i)^-_s|=|F\cap( \gamma+i)|=|F\cap \gamma|+i=\Vert\gamma\Vert_s+i\end{align*} $$

and in a similar way we get that $\lVert \delta \lVert _s+i=\lVert \delta +i\lVert _s$ . The claim follows directly from these equalities.

From the arguments presented in Proposition 5.2, we conclude that $\Delta (e_{\gamma +i},e_{\delta +i})=\Delta (\gamma +i,\delta +i)=l$ for any $i<2n+1$ . This means that the order between $e_{\gamma +i}$ and $e_{\gamma +i}$ is determined by the value of these two functions at l.

Observe that $\rho (\gamma +i,\gamma )\leqslant \rho ^{D_\gamma }=k<l$ and $\rho (\delta +i,\delta )\leqslant \rho ^{D_\delta }=k<l$ for each $i<2n+1$ . In this way,

$$ \begin{align*}0\leqslant \Xi_\gamma(l)=\Xi_{\gamma+j}(l)\not=\Xi_{\delta+j}(l)=\Xi_\delta(l)\geqslant 0.\end{align*} $$

Without any loss of generality let us assume that $j:=\Xi _\gamma (l)>\Xi _\delta (l)=:z$ . Observe that for each i we have that $e_{\gamma +i}(l)<e_{\delta +i}(l)$ if and only if $e_{\gamma +i}(l)=-j$ . Let $C_{even}=\{\lVert \gamma +i\Vert _{l-1}\,:\,i<2n+1\text { is even}\}$ and $C_{\textit {odd}}=\{\{\lVert \gamma +i\Vert _{l-1}\,:\,i<2n+1 \text {is odd}\}$ . Consider the following cases.

Case 1: If $e_\gamma (l)=j$ then $e_{\gamma +i}(l)=j$ for any $i\in C_{even}$ . Since we are assuming that the type is alternating, then $e_{\gamma +1}(l)=-j$ . This implies that $\lVert \gamma +1\Vert _{l-1}>\min (C^{k-1}_j)$ . Consequently, $\Vert \gamma +1\Vert -1=\lVert \gamma \Vert _{l-1}\geqslant \min (C^{k-1}_j)$ . This means that $C_{even}\subseteq C^{l-1}_j$ . However, this is a contradiction since $|C_{even}|=n+1>n\geqslant |C^{l-1}_j|$ .

Case 2: If $e_\gamma (l)=-j$ , we can argue in a similar way to conclude that $C_{\textit {odd}}\subseteq C^{l-1}_j$ . If $|C^{l-1}_j|<n$ , this leads to the same contradiction as in the previous case. On the other hand, if $|C^{l-1}_j|=n$ , then the fact that $e_\gamma (l)=-j$ implies that $\min (C^{l-1}_j)<C_{\textit {odd}}$ . In this way, $n=|C_{\textit {odd}}|<|C^{l-1}_j|-1=n-1$ which is again a contradiction.

By arguing in a similar way but ignoring Case 1, we get the following proposition.

We can use the same ideas as above to prove that the family $\mathbb {E}_{\mathcal {F}}=\langle \Xi _\alpha \rangle _{\alpha \in \omega _1}$ is a subset of $\omega ^\omega $ which is not even $2$ -entangled. Through out the proofs of this section we have been using the fact that $\Delta (o_\alpha ,o_\beta )=\Delta (\alpha ,\beta )=\Delta (e_\alpha ,e_\beta )$ for any two $\alpha ,\beta \in \omega _1$ . Since $\Delta (\alpha ,\beta )=\Delta (\Xi _\alpha ,\Xi _\beta )$ as well (see Lemma 4.5), this means that $\mathbb {E}^{\mathcal {F}}$ is homeomorphic to both $\mathbb {E}_e$ and $\mathbb {E}_o$ via the functions sending $\Xi _\alpha $ to $e_\alpha $ and $o_\alpha $ , respectively. In [Reference Chapital8], we defined a set $\mathbb {E}_\infty $ which is entangled and homeomorphic to $\mathbb {E}^{\mathcal {F}}$ for the exact same reasons. Thus, the results of this section together to the one from [Reference Chapital8] yield following theorem generalizing the one from [Reference Carroy, Levine and Notaro6].

6 More on schemes and entangled sets

As we have mentioned before, the existence of a fully $\Delta $ -capturing scheme with a rapidly growing (scheme) type implies the existence of an entangled set. In particular, this means that the existence of certain fully capturing schemes imply the same result. Thus, it is natural to ask what is the amount of capturing needed in order to build this kind of objects. The main goal of this section is to show that neither the existence of a $2$ - $\Delta $ -capturing schemes or even $2$ -capturing schemes implies the existence of an entangled set. In order to understand the importance of these results it is worth to remark that most of the objects which we know how to construct using schemes just require the scheme to be $2$ -capturing (see [Reference Chapital, González and Todorcevic10]). For example, in [Reference Chapital, González and Todorcevic10] we showed that the existence of a $2$ -capturing scheme implies the failure of BA $(\omega _1)$ . Actually, it is easy to prove that $\{\Xi _\alpha \,:\,\alpha \in \omega _1\}$ is an increasing setFootnote ⁸ by just assuming that $\mathcal {F}$ is $2$ - $\Delta $ -capturing.

When dealing with $2$ - $\Delta $ -capturing schemes, we can actually prove a much more stronger result. This slightly improves the result from [Reference Abraham, Rubin and Shelah1] stating that the existence of an increasing set is compatible with MA.