1 Notation

The notation and terminology used here mostly standard and it follows [Reference Chapital, González and Todorcevic9]. 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 sets X and Y two sets of ordinals, we write $X<Y$ whenever $\max (X)<\min (Y)$ or $X=\emptyset $ . For a set of ordinals X, 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, $R<X\backslash R$ and $X\backslash R<Y\backslash R$ or viceversa for any two $X,Y\in \mathcal {D}$ , we call $\mathcal {D}$ a root-tail-tail $\Delta $ -system. If $(X,<)$ is a partial order and $x\in X$ we let $pred(x)$ to be the set of all immediate predecessors of $X.$ Lastly, if $y\in X$ , we call it successor-like if $pred(x)$ is non-empty, finite, and cofinal inside $(-\infty ,y)=\{x\in X\,:\,x<y\}.$

2 Almost disjoint representations and multiple gaps

Recall that an almost disjoint family (shortly, AD family) over a set X Footnote ¹ is a family $\mathcal {A}$ of infinite subsets of X so that $A\cap B=^*\emptyset $ for any two distinct $A,B\in \mathcal {A}$ .

We say that $\mathcal {A}$ is:

• • n-inseparable if for any pairwise disjoint $\mathcal {L}_0,\dots ,\mathcal {L}_{n-1}\in [\mathcal {A}]^{\omega _1}$ , the tuple $(\mathcal {L}_i\,:\,i<n)$ forms an n-gap. We suppress the index if $n=2$ .

• • $\omega $ -inseparable if $\mathcal {A}$ is n-inseparable for each $n\geqslant 2$ .

• • Jones if for any two disjoint $\mathcal {L}\in [\mathcal {A}]^{\omega }$ and $\mathcal {R}\in [\mathcal {A}]^{\omega _1}$ , $(\mathcal {L},\mathcal {R})$ is not a gap.

The first construction of an inseparable family was done in [Reference Luzin22] by Luzin. A Jones family was implicitly constructed by F. B. Jones in [Reference Jones17]. At first glance, inseparable and Jones seem to be incompatible properties. However, a construction of an inseparable–Jones family was obtained in [Reference Guzmán, Hrušák and Koszmider13] by Guzmán, Hrušák, and Koszmider (building from work by Koszmider in [Reference Koszmider18]).

The classical constructions of both an inseparable AD family and an $(\omega _1,\omega _1)$ -gap share the same basic idea. However, the reader might agree that from these two, the latter is much harder to build. Having this in mind, one might ask whether it is possible to build an $(\omega _1,\omega _1)$ -gap in an easier way by using the elements of an already existing inseparable family as building blocks. A natural attempt of doing this is by starting with an almost disjoint family $\mathcal {A}=\{A_\alpha \,:\,\alpha \in \omega _1\}$ and build a pregap $(\mathcal {L},\mathcal {R})=(L_\alpha ,R_\alpha )_{\alpha \in \omega _1}$ as follows: We start by defining $L_0=A_0$ and $R_0=A_1$ . If $\alpha =\gamma +n$ for some natural number n greater than $0$ , we let $L_\alpha :=L_{\alpha -1}\cup A_{\gamma +2n}$ and $R_\alpha :=R_{\alpha -1}\cup A_{\gamma +2n+1}$ . Assume that at limit steps $\gamma $ we are able to find

$$ \begin{align*}\begin{gathered}(\star) \\\text{two disjoint sets }L\text{ and }R\text{ so that }L_\alpha\subseteq^*L, R_\alpha\subseteq^* R, \text{and}\\ A_{\beta}\cap L=^*\emptyset=^*A_{\beta}\cap R\text{ for every }\alpha<\gamma\text{ each }\gamma\leqslant\beta<\omega_1.\end{gathered}\end{align*} $$

Then we can continue the construction by defining $L_\gamma :=L\cup A_\gamma $ and $R_\gamma :=R\cup A_{\gamma +1}$ . At the end, we will have built a pregap $(\mathcal {L},\mathcal {R})=(L_\alpha ,R_\alpha )_{\alpha \in \omega _1}$ . If $\mathcal {A}$ is inseparable, it is not hard to see that $(\mathcal {L},\mathcal {R})$ will in fact be a gap. So the question is: Is there an inseparable almost disjoint $\mathcal {A}$ for which $(\star )$ holds at every limit step? One can prove that the answer to this question is “yes” under $\mathfrak {b}>\omega _1$ . In fact, under this hypothesis it is possible to carry the construction for any inseparable AD family. Another trivial observation is that any such $\mathcal {A}$ should at least be Jones. For this same reason, we can not expect to show that the construction described above will work for every $\mathcal {A}$ . This is because while it is true that there exists an inseparable Jones family in $ZFC$ , under $\mathfrak {b}=\omega _1$ there is also one inseparable family which is not Jones.

The next definition is based on the discussion presented above. Informally, it captures the idea of being able to embed a given partial order in the power set of a set N using an almost disjoint family over N as “building blocks” of this embedding.

The construction of a gap $(\mathcal {L}, \mathcal {R})$ via an AD family $\mathcal {A}$ described at the beginning of this section would actually yield an AD representation $(\mathcal {L}\cup \mathcal {R},\mathcal {A})$ of the partial order $(2\times \omega _1,\leqslant )$ where $(i,\alpha )\leqslant (j,\beta )$ if and only if $i=j$ and $\alpha \leqslant \beta $ . So now the question is: Is there an inseparable AD family $\mathcal {A}$ which codes $2\times \omega _1$ ? We will answer this question in the affirmative not only for $2\times \omega _1$ but, up to cofinal suborders, for any possible partial order of size $\omega _1$ .

Let us say that a partial order $(X,\leqslant )$ is $\omega _1$ -like if it is well-founded, $|X|=\omega _1$ and $|(-\infty ,x)|\leqslant \omega $ for all $x\in X$ .

We are ready to present the two main theorems regarding almost disjoint families and multiple gaps.

Theorem 2.4 can be stated in a more general way. We will present two new axioms, namely $CA^\Delta $ and $CA^\rho $ , which are consequences of CH and , respectively. These two principles are compatible with arbitrarily large values of the continuum. In particular, they both hold in the Cohen model. Actually, we will show that the existence of an AD family satisfying the conclusion Theorem 2.4 follows from both $CA^\Delta $ and $CA^\rho $ .

The existence of an $\omega $ -inseparable AD family is already interesting as it is the natural generalization to the realm of n-gaps of the well-studied inseparable families. As we show in the following corollary, Theorem 2.4 yields new situations in which $\omega _1$ -n-gaps exist.

Particularly, we get the following corollary. As a direct consequence of it, we also have that there are $\omega _1$ -n-gaps in the Sacks model.

In order to keep illustrating the power of the concept of almost disjoint representations, we now extend the work done by Talayco in [Reference Talayco30]. Specifically, we study the size of groups defined in a similar context as Talayco and prove that, under certain assumptions, these groups are as big as possible.

We say that $(X,\leqslant ,\wedge )$ is a lower semi-lattice if $(X,\leqslant )$ is a partial order and for any $x,y\in X$ , $\inf (x,y)$ exists and its equal to $x\wedge y$ . A family $\mathcal {T}\subseteq \mathscr {P}(\omega )$ is said to be a $*$ -lower semi-lattice if its projection to the Boolean algebra $\mathscr {P}(\omega )/\text {Fin}$ is a lower semi-lattice inside $(\mathscr {P}(\omega )/\text {Fin},\,\subseteq ,\, \cap )$ . That is, for any two $A,B\in \mathcal {T}$ there is $C\in \mathcal {T}$ so that $A\cap B=^*C$ . Note that towers are particular cases of $*$ -lower semi-lattices.

The following lemma is easy. We prove it just to emphasise the reason for defining coherent subsystems just for $*$ -lower semi-lattices.

Lemma 2.9. Let $\mathcal {T}$ be a $*$ -lower semi-lattice and let g be a coherent subsystem of $\mathcal {T}$ . Then $\mathcal {C}(\mathcal {T},g)$ is a pregap. Furthermore, g is non-trivial if and only if $\mathcal {C}(\mathcal {T},g)$ forms a gap.

Proof. Let $A,B\in \mathcal {T}$ . We need to prove that $g(A)\cap (B\backslash g(B))=^*\emptyset .$ For this let $C\in \mathcal {T}$ be so that $C=^*A\cap B$ . Then

$$ \begin{align*}g(A)\cap B=^* (g(A)\cap A)\cap B=^*g(A)\cap C =^*g(C)=^*g(B)\cap C\subseteq ^*g(B). \end{align*} $$

This finishes the argument. Now we will prove the proposed equivalence.

Proof of $\Rightarrow $

We will prove this by contra-positive. Suppose that there is a set C separating the pregap $\mathcal {C}(\mathcal {T},g)$ . For any $A\in \mathcal {T}$ we have that $g(A)\subseteq ^* C$ and $C\cap (A\backslash g(A))=^*\emptyset .$ In this way, $C\cap A=^*g(A)$ . We conclude that C trivializes g.

Proof of $\Leftarrow $

Again, by contra-positive. Suppose that there is C which trivializes g. Then $C\cap A=^*g(A)$ for any $A\in \mathcal {T}$ . In particular $g(A)\subseteq ^* C$ and $C\cap (A\backslash g(A))=\emptyset $ . We conclude that C separates $\mathcal {C}(\mathcal {T},g)$ .

Definition 2.10. Let $\mathcal {T}$ be a $*$ -lower semi-lattice over a set X. For $f,g\in C(\mathcal {T})$ we can define $f\cdot g\in C(\mathcal {T})$ given as

$$ \begin{align*}(f\cdot g)(A)= f(A)\Delta g(A).\end{align*} $$

It is easy to see that $C(\mathcal {T})$ is an abelian (Boolean)Footnote ² group with respect to this operation and $Tr(\mathcal {T})$ is a (normal) subgroup of it. The quotient of these two groups is what we call the gap cohomology group of $\mathcal {T}$ .

Definition 2.11 (Gap cohomology group)

We define the gap cohomology group of $\mathcal {T}$ as the quotient of these two groups. That is,

$$ \begin{align*}G(\mathcal{T})=C(\mathcal{T})/Tr(\mathcal{T}).\end{align*} $$

We say that $f,g\in C(\mathcal {T})$ are cohomologous if the class of f is equal to the class of g inside $G(\mathcal {T})$ , or equivalently, if $f\cdot g\in Tr(\mathcal {T})$ .

In [Reference Talayco30], Talayco defined the gap cohomology group for the particular case of $\omega _1$ -towers. In there he showed that the $2^\omega \leqslant |G(\mathcal {T})|$ for any $\omega _1$ -tower $\mathcal {T}$ . It is worth noting that by that time, Todorčević had already proved (in the 1980’s) that an $(\omega _1, \omega _1)$ -gap can be inserted into any tower (see [Reference Bekkali8] or [Reference Todorcevic34]).

Talayco’s result is based on the following theorem.Footnote ³

He also proved that the $\Diamond $ -principle implies that the size of this group is always $2^{\omega _1}$ . Later, Todorčević noted that the $\aleph _0$ gap theorem already gives you the previous conclusion without assuming any extra axioms (see Theorem 32 in [Reference Talayco31]). Namely, if $2^{\omega }=2^{\omega _1}$ the result is clear due to Talayco’s calculations. On the other hand, if $2^{\omega }<2^{\omega _1}$ then the result follows from an easy counting argument involving the quotient structure of $G(\mathcal {T}).$ In [Reference Morgan25], Morgan constructed an $\omega _1$ -tower whose gap cohomology group can be explicitly calculated. This construction was carried through the use of morasses. Finally, in [Reference Farah12], Farah improved the $\aleph _0$ gap theorem. This was achieved by considering the following concept.

Farah constructed a Haudorff coherent family of functions supported by an $\omega _1$ -tower by forcing it and then appealing to Keisler’s completeness theorem for $L^\omega (Q)$ . Unlike Talayco’s result, the proof idea behind the construction of Farah does not work for any $\omega _1$ -tower. This is due to the nature of Keisler’s theorem.

As an application of the concept of AD representations, we generalize the results of Talayco, Todorčević, Morgan, and Farah regarding the existence of “big gap cohomology groups.” It is worth noting that there is no analogous to the $\aleph _0$ gap theorem for arbitrary $*$ -lower semi-lattices. Namely, if there is a uniformizable AD system (also called strong-Q-sequence), then one can prove that there is a $*$ -lower semi-lattice $\mathcal {T}$ isomorphic to $[\omega _1]^{<\omega }$ for which $G(\mathcal {T})$ is a trivial group. In particular, we can not argue in the same way as Todorčević in order to conclude that the gap cohomology group has always cardinality $2^{\omega _1}$ . Readers who wish to learn more about strong-Q-sequences are referred to [Reference Chodounský10, Reference Steprāns28].

Theorem 2.15. Let $(X,<,\wedge )$ be an $\omega _1$ -like lower semi-lattice. Then there is a $*$ -lower semi-lattice $\mathcal {T}$ isomorphic to X with $|G(\mathcal {T})|=2^{\omega _1}.$

Proof. Consider $X\times 2$ with the order given by $(x,i)<(y,j)$ if and only if $x<y$ and $i=j$ . It is easy to see that $X\times 2$ is also $\omega _1$ -like. In virtue of Theorem 2.3 we can take $\mathcal {T'}=\langle T^{\prime }_{(x,i)}\rangle _{(x,i)\in X\times 2}$ and an inseparable $\mathcal {A}=\langle A_{(x,i)}\rangle _{(x,i)\in X\times 2}$ so that $(\mathcal {T}',\mathcal {A})$ is an almost disjoint representation of $X\times 2$ . For any $x\in X$ let $T_x=T^{\prime }_{(x,0)}\cup T^{\prime }_{(x,1)}. $ It is straightforward that the following properties hold for any $x,y \in X$ :

1. (a) $A_{(x,0)}\cup A_{(x,1)}\subseteq T_x$ .

2. (b) If $y\not \leqslant x$ then $A_{(y,0)}\cup A_{(y,1)}\subseteq ^*T_y\backslash T_x$ . In other words, $(A_{(y,0)}\cup A_{(y,1)})\cap T_x=^*\emptyset $ .

3. (c) $T_x\cap T_y=^*T_{x\wedge y}.$

In particular, properties (b) and (c) imply that $\mathcal {T}=\langle T_x\rangle _{x\in X}$ is a $*$ -lower semi-lattice isomorphic to X. We claim that $|G(\mathcal {T})|=2^{\omega _1}$ . Trivially $|G(\mathcal {T})|\leqslant 2^{\omega _1}$ , so we will only prove the other inequality. For this, let $S\in \mathscr {P}(X)$ . We will build recursively (using that X is well-founded) a coherent subsystem $g_S:\mathcal {T}\longrightarrow \mathscr {P}(\omega )$ such that $A_{(x,0)}\subseteq g_S(T_x)$ for any $x\in X$ and:

$$ \begin{align*}\begin{array}{ccc} \textbf{(A)} &&\textbf{(B)} \\ \text{If }x\in S\text{, then }A_{(x,1)}\cap g_S(T_x)=^*\emptyset. && \text{If }x\not\in S\text{, then }A_{(x,1)}\subseteq g_S(T_x). \end{array} \end{align*} $$

Suppose that $y\in X$ and we have defined $g_S(T_x)$ for any $x<y$ . By means of Lemma 2.9 we know that $C(\mathcal {T}|_y,g_S|_{\mathcal {T}|_{y}})$ is pregap where $\mathcal {T}|_y=\langle T_x\rangle _{x<y}$ . As there are no countable gaps and $|(-\infty ,y)|\leqslant \omega $ , we conclude that there is $C\in \mathscr {P}(\omega )$ separating it. Since $g_S(T_x)\subseteq T_x\subseteq ^*T_y$ for any $x<y$ , we may assume without loss of generality that $C\subseteq T_y$ . Note that $C\cap T_x=^*g_S(T_x)$ for any $x<y$ . We now define $g_S(T_y)$ by cases. If $y\in S$ define $g_S(T_y)$ as $(C\cup A_{(y,0)})\backslash A_{(y,1)}$ . Otherwise define $g_S(T_y)$ as $C\cup A_{(y,0)}\cup A_{(y,1)}$ . Since both $A_{(y,0)}$ and $A_{(y,1)}$ are almost disjoint with $T_x$ for all $x<y$ due to the point (b), it follows that $g_S(T_y)\cap T_x=^* C\cap T_x=^*g_S(T_x)$ for any such x. This finishes the recursion.

Now consider $\mathcal {S}\subseteq [X]^{\omega _1}$ a family of cardinality $2^{\omega _1}$ so that $S\Delta S'$ is uncountable for any two distinct $S,S'\in \mathcal {S}$ . The key fact needed to finish the proof is that if $x\in S\Delta S'$ then $A_{(x,1)}\subseteq g_S\cdot g_{S'}(T_x)$ and $A_{(x,0)}\subseteq T_x\backslash (g_S\cdot g_{S'}(T_x))$ . In this way, any separation of the pregap $\mathcal {G}(\mathcal {T},g_S\cdot g_{S'})$ would also separate the pregap $(A_{(x,1)},A_{(x,0)})_{x\in S\Delta S'}$ which is impossible since $\mathcal {A}$ is an inseparable family and $S\Delta S'$ is uncountable. This shows that $g_S$ and $g_{S'}$ are not cohomologous. Since $|S|=2^{\omega _1}$ , then $|G(\mathcal {T})|=2^{\omega _1}$ .

3 Finitizations of and CH from construction schemes

Guessing principles are among the first things which come to mind when speaking about combinatorics over $\omega _1$ . Some of the most utilized guessing principles are Jensen’s $\Diamond $ , the Continuum Hypothesis, and Ostszewski’s . These three axioms can be formulated in order to assert the existence of a sequence $\langle C_\gamma \rangle _{\gamma \in \text {Lim}}$ (where $\text {Lim}$ denotes the limit ordinals below $\omega _1$ ) of countable subsets of $\omega _1$ such that $C_\gamma \subseteq \gamma $ for each $\gamma $ , and such that for any $S\in [\omega _1]^{\omega _1}$ and $\beta \in \omega _1$ , the following property hold:

1. (Under ◊) $\{\gamma \in \text {Lim}\,:\,\gamma \cap S=C_\gamma \text { and }\sup (C_\gamma )=\gamma \}$ is stationary. A sequence satisfying this property its called a $\Diamond $ -sequence.

2. (Under ) $\{\gamma \in \text {Lim}\,:\,C_\gamma \subseteq S\text { and }\sup (C_\gamma )=\gamma \}$ is stationary. A sequence satisfying this property its called a -sequence.

3. (Under CH) $\{\gamma \in \text {Lim}\,:\,C_\gamma \sqsubseteq S\text { and }\beta \leqslant \sup (C_\gamma )\}$ is stationary.

It can be seen at a first glance that these principles are related. For example, given a sequence $\langle C_\gamma \rangle _{\gamma \in \text {Lim}}$ , the intersection of the two latter sets described above is equal the former one when we take $\beta =0$ . From this fact, it already follows that $\Diamond $ implies both CH and . It is not completely direct that ${+}\,CH$ imply $\Diamond $ , even if there is one single sequence testifying these two axioms. Nevertheless, such implication indeed hold, and in some sense, this tells us that the interplay between these axioms goes as far as possible.

In [Reference Todorcevic35], Todorčević introduced a generalization of Jensen’s $(\omega ,1)$ -gap morasses which he called construction schemes. These objects are special families of finite subsets of $\omega _1$ which serve as a tool for building uncountable objects by means of finite approximations. In that same paper, he introduced the capturing axiom $FCA$ . Such axiom states the existence of construction schemes with certain guessing properties which can be interpreted as finite dimensional versions of the properties possessed by a $\Diamond $ -sequence. In fact, it turns out that $FCA(part)$ is implied by $\Diamond $ . As it has been proved in papers, such as [Reference Chapital, González and Todorcevic9, Reference Hida and Koszmider15, Reference Lopez19–Reference Lopez and Todorcevic21, Reference Todorcevic35], this capturing axiom is sufficient to imply the existence of many of the most well-known and studied objects whose existence already follows from Jensen’s principle. There are variations of the axiom $FCA(part)$ called $CA_n(part)$ (for each natural number $2\leqslant n$ ) and $CA(part)$ as well as their non “part” variants $FCA$ , $CA_n$ , and $CA$ whose hierarchy is shown in the following diagram:

The arrows in this diagram are not reversible. Hence, this collection of axioms seems to be a good framework for studying the strength of the consequences of $\Diamond $ . Of course, there are already other kinds of axioms which serve a similar purpose. For example, the parametrized $\Diamond $ -principles introduced in [Reference Moore, Hrušák and Amonja24] by Džamonja, Hrušák, and Moore.

The main goal of this section is to describe finite dimensional versions of and CH, called $CA^\rho $ and $CA^\Delta $ (and their respective variants), which play an analogous roll in relation to the capturing axioms for construction schemes as the one played by their “infinite dimensional” counterparts with respect to the $\Diamond $ -principle.

We assume that the general audience is yet not familiar with the theory of construction schemes. For that reason, we will dedicate the first part of this section to recall some of the basic definitions and results regarding this theory. After doing so, we will be in the position to state the axioms $CA^\rho $ and $CA^\Delta $ in a precise way.

A 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 $ :

$$ \begin{align*} \begin{array}{lll} (a)~m_0=1,&& (c)~m_k>r_{k+1}, \\ (b)~n_{k+1}\geqslant 2, && (d)~m_{k+1}=r_{k+1}+(m_k-r_{k+1})n_{k+1}. \end{array} \end{align*} $$

We say that type is good if for each $r\in \omega $ , there are infinitely many k’s for which $r=r_k$ . Throughout this text, we will only work with good types. We say that partition of $\omega $ , namely $\mathcal {P}$ , is compatible with $\tau $ if $\tau $ is good when restricted to each member of $\mathcal {P}$ . That is, for each $P\in \mathcal {P}$ and every $r\in \omega $ there are infinitely many $k\in P$ for which $r_k=r$ .

Given a set X of ordinals, a construction scheme (of type $\tau $ ) over X is a family $\mathcal {F}\subseteq \mathrm {Fin}(X)$ which: Is cofinal in $(\mathrm {Fin}(X),\subseteq )$ , 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 $\Delta $ -system with root $R(F)$ such that $|R(F)|=r_{k+1}$ and $R(F)<F_0\backslash R(F)<\dots < F_{n_{k+1}-1}\backslash R(F).$

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 $a_i=r_{k+1}+i\cdot (m_{k+1}-r_{k+1})$ . 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. If $\mathcal {F}$ is a construction scheme over X, we call X the domain of $\mathcal {F}$ and we write it as $dom(\mathcal {F})$ . In practice, we are only interested in construction schemes over $\omega _1$ . However, in order to build such schemes it is convenient to work also with schemes with both finite and countable domains. We will frequently use the following fact.

Each construction scheme $\mathcal {F}$ is associated with some natural functions of countable codomain. The first of such functions is the one derived from the cofinality condition in the definition of a scheme. Namely, given $\alpha ,\beta \in dom(\mathcal {F})$ there is $F\in \mathcal {F}$ such that $\{\alpha ,\beta \}\subseteq F$ . Hence, we can define $\rho :dom(\mathcal {F})^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 $A\subseteq dom(\mathcal {F})$ , 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 $\alpha \in dom(\mathcal {F})$ 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 is a useful fact that for any $k\in \omega $ (with $m_k\leqslant |dom(\mathcal {F})|$ ) 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 $\alpha \in dom(\mathcal {F})$ , the k-cardinality function $\lVert \alpha \rVert _{\_}:\omega \longrightarrow \omega $ is defined as

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

It is helpful to remember that if $F\in \mathcal {F}_k$ is such that $\alpha \in F$ , then $F(\lVert \alpha \rVert _k)=\alpha $ . Recall that if $f,g:\omega \longrightarrow \omega $ are distinct, we can define $\Delta (f,g):=\min (\,k\in \omega \,:\,f(k)\not =g(k)\,)$ . We put $\Delta (f,g):=\omega $ whenever $f=g$ . In the case of construction schemes, this leads to the definition of the function $\Delta :dom(\mathcal {F})^2\longrightarrow \omega +1$ which is defined as

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

In the following lemma, we describe the most basic properties of this function as well as its relation with the $\rho $ -function.

The last canonical function that we will present is the one which is related to the property $(ii)$ in the definition of a scheme. Namely, given $\alpha \in dom(\mathcal {F})$ we define the function $\Xi _\alpha :\omega \longrightarrow \{-1\}\cup \omega $ as follows; If $1\leqslant k\in \omega $ and $m_k\leqslant |X|$ , then there is $F\in \mathcal {F}_k$ so that $\alpha \in F$ . According to the 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 either $k=0$ or $m_k>|X|$ , we simply define $\Xi _\alpha (k)$ as $0$ .

The next lemma relates the functions $\rho $ , $\Xi $ , and $\Delta $ in a very nice way. In fact, this is, by experience, the most utilized result regarding the basic theory of construction schemes.

As suggested in the previous lemma, there is not much to be said about the functions $\Xi _\alpha $ and $\Xi _\beta $ for the k’s which live in the open interval $(\Delta (\alpha ,\beta ),\rho (\alpha ,\beta ))$ . On the other hand, the relation between $\Xi _\alpha $ and $\Xi _\beta $ can be fully described for k’s outside the afforded mentioned interval. At this point, the notion of capturing comes into play. Naively speaking, a construction scheme over $\omega _1$ is capturing (up to some degree) if for any uncountable subset S of $\omega _1$ there are distinct $\alpha ,\beta $ for which the gap between $\Delta (\alpha ,\beta )$ and $\rho (\alpha ,\beta )$ is non-existent. The importance of this property is that, in this case, Lemma 3.5 can be used at its full power.

The original definition of capturing introduced in [Reference Todorcevic35] is based on the notion of a captured $\Delta $ -system stated below.

A construction scheme $\mathcal {F}$ over $\omega _1$ is then called n-capturing (fully capturing) if every uncountable family of finite subsets of $\omega _1$ contains a captured (fully captured) $\Delta $ -system of size n. If $\mathcal {F}$ is n-capturing for each $n<\omega $ it is called both $\omega $ -capturing and capturing.

The new capturing axioms associated with and CH are defined by weakening the second item of Definition 3.6 in two natural ways. Namely, by deleting either the first or last term in the chain of equalities appearing in such item. To motivate further the analogy between the already studied capturing axioms and the ones which we study in this manuscript, we will present them in a way which reassembles the presentation of $\Diamond $ , , and CH given at the beginning of this section.

Definition 3.7 ( $\ast $ -Captured $\Delta $ -systems)

Consider $\mathcal {F}$ a construction scheme, $X=dom(\mathcal {F})$ , and $\ast $ a family of functions from $X^2$ to $\omega $ . Let $1\leqslant n$ , $m\in \omega $ , and $\mathcal {D}=\langle D_i\rangle _{i<n}\subseteq [X]^{m}$ be a root-tail-tail $\Delta $ -system with a root R of cardinality r. Given $l\in \omega $ , we will say that $\mathcal {D}$ is $\ast $ -captured at level l if

$$ \begin{align*} \begin{gathered}\textbf{(I)}\\ \text{For all }i<n\text{ and each }a<m, \\ \Xi_{D_i(a)}(l)=\begin{cases}-1&\text{if }a< r\\ i&\text{if }a\geqslant r \end{cases} \end{gathered} \end{align*} $$

and for every $\nu \in \ast $ the following condition hold:

$$ \begin{align*}\begin{gathered} \textbf{(II}_\nu\textbf{)} \\ \text{For all }i<j<n\text{ and }r\leqslant a<m,\\ \nu(D_i(a),D_j(a))=l. \end{gathered} \end{align*} $$

We denote the family of all $\ast $ -captured $\Delta $ -systems at level l as $\text {Cap}^\ast _l(\mathcal {F})$ . Finally, if $\mathcal {D}$ is $\ast $ -captured at level l and $n=n_l$ , we will say that $\mathcal {D}$ is $\ast $ -fully captured. The family of all $\Delta $ -systems which are $\ast $ -fully captured at level l will be denoted as $\text {FCap}^\ast _l(\mathcal {F})$ .

Suppose that $\mathcal {F}$ and $\ast $ are as in the previous definition and that $\omega _1=dom(\mathcal {F})$ . We say that $\mathcal {F}$ is $(\mathcal {P},n)$ - $\ast $ -capturing (for some $n\in \omega $ and a partition $\mathcal {P}$ of $\omega $ ) if for any uncountable $\mathcal {S}\subseteq \mathrm {Fin}(\omega _1)$ and each $P\in \mathcal {P}$ , the set

$$ \begin{align*}\{l\in P\,:\,[\mathcal{S}]^n\cap \text{Cap}^\ast_l(\mathcal{F})\not=\emptyset\}\end{align*} $$

is infinite. If $\mathcal {F}$ is $(\mathcal {P},n)$ - $\ast $ -capturing for any $2\leqslant n \in \omega $ , we will say that it is $(\mathcal {P},\omega )$ - $\ast $ -capturing (or simply, $\mathcal {P}$ - $\ast $ -capturing). Finally, we will say that $\mathcal {F}$ is fully $\mathcal {P}$ - $\ast $ -capturing if for any uncountable $\mathcal {S}\subseteq \mathrm {Fin}(\omega _1)$ and $P\in \mathcal {P}$ , the set $\{ l\in P\,:\,\mathrm {Fin}(S)\cap \text {FCap}^\ast _l(\mathcal {F})\not =\emptyset \}$ is infinite. Whenever $\mathcal {P}$ is the trivial partition $\{\omega \}$ , we will omit it from the notation.

By now, we have presented all the necessary information needed to state the axioms which gave rise to this manuscript. In order to avoid unnecessary verbatim, let us first fix $\ast \in \{\,\{\Delta \},\{\rho \},\{\Delta ,\rho \}\,\}$ :

Fully $*$ -capturing axiom [FCA $^\ast $ ]: There is a fully $\ast $ -capturing construction scheme over $\omega _1$ of every possible good type.

Fully capturing Aaxiom with partitions [FCA $^\ast $ (part)]: There is a fully $\mathcal {P}$ - $\ast $ -capturing construction scheme over $\omega _1$ for every good type $\tau $ and each partition $\mathcal {P}$ compatible with $\tau $ .

n - $\ast $ -Capturing axiom [CA $^\ast _n$ ]: For any $n'\leqslant n$ , there is an $n'$ - $\ast $ -capturing construction scheme over $\omega _1$ of every possible good type satisfying that $n'\leqslant n_k$ for each $k\in \omega \backslash 1$ .

n - $\ast $ -Capturing axiom with partitions [CA $^\ast _n$ (part)]: For any $n'\leqslant n$ , there is a $(\mathcal {P},n')$ - $\ast $ -capturing construction scheme over $\omega _1$ for every good type $\tau $ satisfying that $n'\leqslant n_k$ for each $k\in \omega \backslash 1$ and each partition $\mathcal {P}$ compatible with $\tau $ .

$\ast $ -Capturing axiom [CA $^\ast $ ]: CA $^\ast _n$ holds for each $n\in \omega $ and there is a $\ast $ -capturing construction scheme over $\omega _1$ for every good type satisfying that the sequence $\langle n_{k+1}\rangle _{k\in \omega }$ is non-decreasing and unbounded.

Capturing axiom with partitions [CA $^\ast $ (part)]: CA $^\ast _n$ (part) holds for each $n\in \omega $ and there is a $ \mathcal {P}$ - $\ast $ -capturing construction scheme over $\omega _1$ for every good type $\tau $ satisfying that the sequence $\langle n_{k+1}\rangle _{k\in \omega }$ is non-decreasing and unbounded and each partition $\mathcal {P}$ compatible with $\tau $ .

The two following theorems will be proved in Sections 7 and 8, respectively.

It will be clear that simple modifications of the proofs of the theorems above, actually yield that $FCA^\Delta (part)$ and $CA^\rho (part)$ follow from CH and , respectively.

We finish this section with one last remark. Suppose that $\mathcal {F}$ is a scheme with domain $\omega _1$ , and that $\ast $ is as in Definition 3.7. We say that $\mathcal {F}$ is weakly n- $\ast $ -capturing (for some $n\in \omega $ ) if for any uncountable $\mathcal {S}\subseteq [\omega _1]^1$ , the set

$$ \begin{align*}\{l\in P\,:\,[\mathcal{S}]^n\cap \text{Cap}^\ast_l(\mathcal{F})\not=\emptyset\}\end{align*} $$

is non-empty. The concepts of weakly $\ast $ -capturing and weakly-fully $\ast $ -capturing as well as their partition variations have the expected meaning.

In [Reference Todorcevic35], it was proved that the notions of capturing and weakly capturing schemes are equivalent.

Clearly every $\ast $ -capturing notion implies its weakly- $\ast $ -capturing variant. However, the reverse implication is, in general, not true. For example, if $\ast =\{\Delta \}$ . On one hand, the existence of a $2$ - $\Delta $ -capturing contradicts Baumgartner’s axiom (Section 5). On the other hand, weakly $2$ - $\Delta $ -capturing schemes exist in ZFC alone.

4 Proof of Theorem 2.4

In this section we will prove Theorem 2.4, namely, that if either or CH hold, then there is an $\omega $ -inseparable $AD$ family which codes every $\omega _1$ -like order. What we will actually do is to prove via the Theorems 4.2 and 4.5 that if there is either an n- $\Delta $ -capturing or n- $\rho $ -capturing construction scheme, then there is an n-inseparable family which codes any $\omega _1$ -like order (here, $n\in \omega +1$ ).

The following result essentially due to Hausdorff.

Theorem 4.2. Let $\mathcal {F}$ be a scheme of type $\langle m_k,n_{k+1},r_{k+1}\rangle _{k\in \omega }$ and $3\leqslant n\in \omega $ . Let $N_0=\emptyset $ and given $0<k\in \omega $ , consider $N_k= m_{k-1}^{n_k}\times \{k\}$ and $N=\bigcup \limits _{k\in \omega } N_k$ . Given $\alpha \in \omega _1$ and $k\in \omega $ we define $A^k_\alpha =\emptyset $ if $k=0$ , and if $k>0$ we let

$$ \begin{align*}A^k_\alpha:=\{(\sigma,k)\in N_k\,:\,\Xi_\alpha(k)\geqslant 0\text{ and }\sigma(\Xi_\alpha(k))=\lVert \alpha\rVert_{k-1}\}.\end{align*} $$

Finally, let $A_\alpha :=\bigcup \limits _{k\in \omega } A^k_\alpha $ . Then $\mathcal {A}_{\mathcal {F}}=\langle A_\alpha \rangle _{\alpha \in \omega _1}$ is an inseparable $AD$ -family. Furthermore, if either $\mathcal {F}$ is n- $\rho $ -capturing or n- $\Delta $ -capturing, then $\mathcal {A}_{\mathcal {F}}$ is an n-inseparable. In particular, if $\mathcal {F}$ is either $\rho $ -capturing or $\Delta $ -capturing, then $\mathcal {A}_{\mathcal {F}}$ is $\omega $ -inseparable.

Proof. Each $A_\alpha $ is infinite because $\Xi _\alpha (k)\geqslant 0$ for infinitely many k’s. It is also easy to see that for distinct $\alpha $ and $\beta $ we have that

$$ \begin{align*}A_\alpha\cap A_\beta\subseteq \bigcup\limits_{k\leqslant \rho(\alpha,\beta)} (A_\alpha^k\cap A_\beta^k). \end{align*} $$

Therefore, $A_{\mathcal {F}}$ is an AD family. Before proving the second part note that if $D\in [\omega _1]^n$ and $0\leqslant k\in \omega $ are such that $\Xi _{D(i)}(k)=i$ for each i, then

$$ \begin{align*}\{(\sigma,k)\in N_k\,:\,\forall i<n\,( \sigma(i)=\lVert D(i)\rVert_{k-1})\}\subseteq \bigcap\limits_{i<n} A^k_{D(i)}. \end{align*} $$

So in particular, $\big (\bigcap \limits _{i<n} A^k_{D(i)}\big )\cap N_k\not =\emptyset $ .

We proceed to prove that $\mathcal {A}_{\mathcal {F}}$ is inseparable. Indeed, given $\alpha \in \omega _1$ and $n\in \omega $ , we have that $\{\xi <\alpha \,:\,A_\xi \cap A_\alpha \subseteq \bigcup \limits _{k\leqslant n} N_k\}\subseteq (\alpha )_n$ due to Lemma 3.5. In particular, this means that the set on the left is finite. This proves (modulo a bijection) that the hypotheses of Lemma 4.1 hold, so we are done.

Assume that $\mathcal {F}$ is n- $\rho $ -capturing (resp. n- $\Delta $ -capturing) and let $\langle I_i\rangle _{i<n}$ be a sequence of uncountable disjoint subsets of $\omega _1$ . We shall prove that $( \langle A_\alpha \rangle _{\alpha \in I_i}\,:\,i<n)$ is an n-gap. Assume towards a contradiction that this is not the case and let $c:n\longrightarrow \mathscr {P}(N)$ be a witness of this fact. By refining each $I_i$ , we may assume without loss of generality that for each $i<n$ there is $k_i\in \omega $ so that $A_\alpha ^k\subseteq c(i)$ for all $\alpha \in I_i$ and $k\geqslant k_i$ . As $\bigcap \limits _{i<n} c(i)$ is finite, there is $k>\max (k_i\,:i<n)$ for which this intersection is disjoint from $N_l$ for each $l\geqslant k$ . Now, let $\mathcal {D}\subseteq [\omega _1]^n$ be an uncountable family of pairwise disjoint sets so that $D(i)\in I_i$ for all $D\in \mathcal {D}$ and $i<n$ . Since $\mathcal {F}$ is n- $\rho $ -capturing (resp. n- $\Delta $ -capturing) there exists a subfamily $\langle D_i\rangle _{i<n}$ of $\mathcal {D}$ and $l>k$ for which $\langle D_i\rangle _{i<n}$ is $\rho $ -captured (resp. $\Delta $ -captured) at level l. Since the $D_i$ ’s are pairwise disjoint, we have that $\Xi _{D_i(i)}(l)=i$ for each $i<n$ . Therefore, $\emptyset \not =\big (\bigcap \limits _{i<n}A^l_{D_i(i)} \big )\cap N_l\subseteq \big (\bigcap \limits _{i<n} c(i)\big )\cap N_l$ . This contradiction finishes the proof.

Now, we will prove that $\mathcal {A}_{\mathcal {F}}$ codes any $\omega _1$ -like order. For that, we need the following easy lemma.

Let $(X,\leqslant )\ \omega _1$ -like and $\phi :X\longrightarrow \omega _1$ as above. In the following lemma, in some sense, we will pull back a construction scheme from $\omega _1$ to some sort of “construction scheme” on X. This suggests that the theory of construction schemes could be generalized to other partial orders of size $\omega _1$ . The extend and utility of this idea remains mainly unexplored.

Lemma 4.4. Let $\mathcal {F}$ be a scheme and let X and $\phi $ be as in Lemma 4.3. For each $x\in X$ and $k\in \omega $ define $M^k_x=\{\, z\leqslant x\,:\, \phi (z)\in (\phi (x))_k\,\}$ . The following properties hold for each $x,y\in X$ and $k\in \omega $ :

1. (1) If $\inf (x,y)$ exists and $k>\rho ^{\phi [\{x,y,\inf (x,y)\}]}$ , then $M^k_x\cap M^k_y=M^k_{\inf (x,y)}.$

2. (3/2) If $y\leqslant x$ and $k>\rho ^{\phi [\{x,y\}]}$ , then $M^k_y\subseteq M^k_x$ .

3. (2) If $y\not \leqslant x$ , then $y\in M^k_y\backslash M^k_x.$

4. (3) If x is successor-like and $k>\rho ^{\phi [pred(x)\cup \{x\}]}$ , then $M^k_x\backslash \big ( \bigcup \limits _{z\in pred(x)} M^k_z \big )=\{x\}$ .

5. (4) If x and y are incompatible, then $M^k_x\cap M^k_y=\emptyset .$

Proof. The points (2) and (4) are trivial and the point ( $\frac {3}{2}$ ) follows directly from (1). Therefore, we will only prove (1) and (3).

Proof of $(1)$

Since $\phi $ preserves the order then $\phi (\inf (x,y))\leqslant \min (\phi (x),\phi (y))$ which implies that $(\phi (\inf (x,y))_k\sqsubseteq (\phi (x))_k\cap (\phi (y))_k$ . In this way, if $z\in M^k_{\inf (x,y)}$ then $z\leqslant x,y$ by definition of the infimum, and z belongs to both $(\phi (x))_k$ and $(\phi (y))_k$ because $z\in (\phi (\inf (x,y)))_k$ . Therefore $z\in M^k_x\cap M^k_y$ . On the other hand, if $z\in M^k_x\cap M^k_y$ then $z\leqslant x$ and $z\leqslant y$ , so $z\leqslant \inf (x,y)$ . Furthermore, $\phi (z)\in (\phi (x))_k\cap (\phi (y))_k$ and $\phi (z)\leqslant \phi (\inf (x,y))$ . As $(\phi (\inf (x,y)))_k$ is an initial segment of such intersection then $\phi (z)$ belongs to it. Consequently $z\in M^k_{\inf (x,y)}$ .

Proof of $(3)$

Note that since $x\not \leqslant z$ for any $z\in pred(x)$ then $x\in M^k_x\backslash M^k_z$ for any such z. This proves the inclusion from right to left. To show that the one from left to right also holds, let

$$ \begin{align*}w\in M^k_x\backslash\big( \bigcup\limits_{z\in pred(x)} M^k_z \big).\end{align*} $$

Suppose towards a contradiction that $w<x$ and consider $z\in pred(x)$ with $w\leqslant z$ . Since $k>\rho (\phi (x),\phi (z))$ we conclude that $(\phi (z))_k\sqsubseteq (\phi (x))_k$ . But then $\phi (w)\in (\phi (z))_k$ because $\phi (w)\leqslant \phi (z)$ and $\phi (w)\in (\phi (x))_k$ . Consequently $w\in M^k_z$ which is a contradiction. This finishes the proof.

Theorem 4.5. Let $\mathcal {A}=\mathcal {A}_{\mathcal {F}}$ be the $AD$ family constructed in Theorem 4.2 and let $(X,\leqslant )$ an $\omega _1$ -like order. Then there is a family $\mathcal {T}=\langle T_x\rangle _{x\in X}\subseteq [\omega ]^{\omega }$ and a re-indexing of $\mathcal {A}$ as $\langle \hat {A}_x\rangle _{x\in X}$ so that $(\mathcal {T},\mathcal {A})$ is an almost disjoint representation of X.

Proof. Let $\phi :X\longrightarrow \omega _1$ be as in Lemma 4.3 and $ M^k_x$ be as in Lemma 4.4 for any $x\in X$ and $k\in \omega $ . Fix $x\in X$ and let $\hat {A}_x=A_{\phi (x)}.$ This defines the re-indexing of $\mathcal {A}$ . Now, for each $k\in \omega $ and $x\in X$ , let

$$ \begin{align*}T^k_x=\bigcup \{\,A_{\phi(z)}^{k+1}\,:\,z\in M^k_x\,\}=\bigcup\{\,A^{k+1}_\xi\,:\,\xi\in \phi[M^k_x]\,\}\end{align*} $$

$$ \begin{align*}T_x=\bigcup\limits_{k\in\omega}T^k_x.\end{align*} $$

We claim that $(\mathcal {T},\mathcal {A})$ is an almost disjoint representation of X. In the following paragraphs we will show that the points $(a)$ – $(e)$ of Definition 2.1 are satisfied for such pair.

Proof of $(a)$

Given $k>0$ we have that $A^k_{\phi (x)}\subseteq T^{k-1}_x$ because x is always an element of $M^k_x$ . Therefore $\hat {A}_x=A_{\phi (x)}\subseteq T_x$ .

Before proving the remaining points let us fix some notation. Given $P\in [\omega _1]^{<\omega }$ and $k\in \omega $ , let $A^k_P=\bigcup \{A^k_\alpha \,:\,\alpha \in P\}$ . Note that if $P,Q\in [\omega _1]^{<\omega }$ and $k>\rho ^{P\cup Q}$ , then we have the following:

$$ \begin{align*}A^k_P\cap A^k_Q&=A^k_{P\cap Q},\\A^k_P\cup A^k_Q&=A^k_{P\cup Q},\\A^k_{P}\backslash A^k_Q&=A^k_{P\backslash Q}.\end{align*} $$

We will apply these three equalities for finite sets of the form $\phi [M^k_x]$ . More precisely, observe that if $x,y\in X$ and $k>\rho (\phi (x),\phi (y))$ then $\rho ^{\phi [M^k_x]\cup \phi [M^k_y]}<k+1$ . This is because if we take $F\in \mathcal {F}_k$ with $\phi (x),\phi (y)\in F$ , then $\phi [M^k_x]\cup \phi [M^k_y]=\phi [M^k_x\cup M^k_y]\subseteq (\phi (x))_k\cup (\phi (y))_k\subseteq F.$

Proof of $(c)$

Let $x,y\in X$ and suppose that $\inf (x,y)$ exists. Take an arbitrary $k>\rho ^{\phi [\{x,y,\inf (x,y)\}]}$ . By means of the point $(1)$ of Lemma 4.4, $M^k_x\cap M^k_y=M^k_{\inf (x,y)}$ . In this way,

$$ \begin{align*} T^k_x\cap T^k_y =A^{k+1}_{M^k_x}\cap A^{k+1}_{M^k_y} =A^{k+1}_{M^k_x\cap M^k_y} =A^{k+1}_{M^k_{\inf(x,y)}}=T^k_{\inf(x,y)}. \end{align*} $$

Therefore, $T_x\cap T_y\hspace{-1pt}=\hspace{-1pt}\bigcup \limits _{k\in \omega } \big ( T^k_x\cap T^k_y\big )\hspace{-1pt}=^*\hspace{-1pt}\bigcup \{T^k_{\inf (x,y)}\,:\, k>\rho ^{\phi [x,y,\inf (x,y)]}\}\hspace{-1pt}=^*$ $T_{\inf (x,y)}.$

The remaining parts of the theorem are proved in a completely similar way. Because of this, we leave the calculations to the reader.

5 Entangled sets and $\sigma $ -monotone spaces

One of Cantor’s most famous theorems is that any two countable dense total orders without endpoint are isomorphic. Particularly, this result can be applied to any two countable dense subsets of the reals. Thus, it is natural to ask whether Cantor’s theorem can be extended to higher infinities (inside $\mathbb {R}$ ).

In [Reference Baumgartner7], Baumgartner proved that it is consistent that any two $\omega _1$ -dense sets of reals are isomorphic. This assertion is now known as the Baumgartner’s axiom for $\omega _1$ . More generally, if $\kappa $ is an infinite cardinal then we can define:

Baumgartner’s axiom for $\kappa $ [ $BA(\kappa )$ ]: Any two $\kappa $ -dense sets of reals are isomorphic.

One of the ways of proving Cantor’s theorem is by defining the forcing of finite approximations of isomorphisms and making use of the Sikorski’s lemma. Hence, it is natural to think that maybe $BA(\kappa )$ has some relation with $MA$ . However, this is not the case. In [Reference Avraham and Shelah6], Abraham and Shelah showed that $MA$ does not imply $BA(\omega _1)$ . For that, they introduced the objects which we will study in this section. Namely, the increasing and entangled sets. Readers interested in learning more about them may also look at [Reference Chodounský and Zapletal11, Reference Guzmán and Todorcevic14, Reference Miyamoto and Yorioka23, Reference Todorčević32, Reference Todorčević33, Reference Yuasa36].

Definition 5.2 (Realization)

Let $k\in \omega $ and $t:k\longrightarrow \{<,>\}$ . Given a total order $(X,<)$ and $a,b\in [X]^k$ , we say that $(a,b)$ realizes t if

$$ \begin{align*}a(i)\:t(i)\:b(i)\end{align*} $$

for each $i<k.$ By $T(a,b)$ we denote the unique t which is realized by $(a,b).$

Given $n\in \omega $ , let $inc_n:n\longrightarrow \{<,>\}$ be the constant function with value $<$ .

Lemma 5.4. Let $(X,<)$ be a total order, $k\in \omega $ and $\mathcal {E}\in [X]^{\omega _1}$ injectively enumerated as $\langle r_\alpha \rangle _{\alpha \in \omega _1}$ . Then $\mathcal {E}$ is k-entangled if and only if for every uncountable family $\mathcal {C}\subseteq [\omega _1]^{k}$ of pairwise disjoint sets and each $t:k\longrightarrow \{<,>\}$ there are distinct $c,d\in \mathcal {C}$ for which

$$ \begin{align*}r_{c(i)}\,t(i)\,r_{d(i)}\end{align*} $$

for each $i<k.$

Proof. As both implications are proved in a completely similar way, we will only show the one from left to right. For this purpose, suppose that $\mathcal {E}$ is k-entangled. Let $\mathcal {C}\subseteq [\omega _1]^k$ be an uncountable family of pairwise disjoint sets and $t:k\longrightarrow \{<,>\}$ . Given $c\in \mathcal {C}$ , take $h_c:k\longrightarrow k$ the unique function satisfying that $i<j$ if and only if $r_{c(h(i))}<r_{c(h_c(j))}$ for every $i,j<k$ . By refining $\mathcal {C}$ , we can suppose without loss of generality that there is h for which $h=h_c$ for all $c\in \mathcal {C}$ . The key observation is that h codes the increasing enumeration (with respect to order in X) of $\langle r_{c(i)}\rangle _{i<k}$ whenever $c\in \mathcal {C}$ . That is, $r_{c(h(0))}<\dots <r_{c(h(k-1))}$ . Now, let $t'=t\circ h$ . By the previous observation and since $\mathcal {E}$ is k-entangled, there are distinct $c,d\in \mathcal {C}$ for which

$$ \begin{align*}T(\langle\, r_{c(h(i))}\,\rangle_{i<k},\langle\, r_{d(h(i))}\,\rangle_{i<k})=t'.\end{align*} $$

To finish, take an arbitrary $i<k$ and let $i'=h^{-1}(i)$ . Then $r_{c(h(i'))}\,t'(i')\, r_{d(h(i'))}$ . But $r_{c(h(i'))}=r_{c(i)}$ , $t'(i')=t(i)$ and $r_{d(h(i'))}=r_{d(i)}$ . In this way, $r_{c(i)}\,t(i)\, r_{d(i)}$ .

Theorem 5.7 (Under FCA $^\Delta $ )

There is an entangled set.

Proof. Let $\mathcal {F}$ be a fully $\Delta $ -capturing scheme of type $\langle m_k,n_{k+1},r_{k+1}\rangle _{k\in \omega }$ satisfying that $n_{k+1}\geqslant 2^{m_k}+1$ for each $k\in \omega .$

Given $k\in \omega $ , let us enumerate $\mathscr {P}(m_{k}\backslash r_{k+1})$ , possibly with repetitions, as $\langle C^k_i\rangle _{0<i<n_{k+1}}$ Now, for each $\alpha \in \omega _1$ let $f_\alpha :\omega \longrightarrow \mathbb {Z}$ be given as

$$ \begin{align*}f_\alpha(k)=\begin{cases}0 &\textit{if }k=0 \textit{or } \Xi_\alpha(k)=-1\\ \Xi_\alpha(k)&\textit{if }k>0,\,\Xi_\alpha(k)\geqslant 0 \textit{ and }\lVert\alpha\rVert_{k-1}\in C^{k-1}_{\Xi_\alpha(k)}\\ -\Xi_\alpha(k)&\textit{ if }k>0,\,\Xi_\alpha(k)\geqslant 0 \textit{and }\rVert\alpha\lVert_{k-1}\,\notin C^{k-1}_{\Xi_\alpha(k)}.\\ \end{cases}\end{align*} $$

Let $\mathcal {E}=\langle f_\alpha \rangle _{\alpha \in \omega _1}$ . We claim that $\mathcal {E}$ is an entangled set in $(\mathbb {Z}^\omega ,<_{lex})$ . Indeed, let $k\in \omega $ , $t:k\longrightarrow \{>,<\}$ and $\mathcal {C}\subseteq [\omega _1]^k$ be an uncountable family of pairwise disjoint sets. As $\mathcal {F}$ is fully $\Delta $ -capturing, there is $\{c_0,\dots ,c_{n_{l}-1}\}\in [\mathcal {C}]^{n_l}$ which is $\Delta $ -captured at some level $l>0$ . We claim that there is $i<n_l$ so that $T(c_0,c_i)=t$ . First of all, note that $\Delta (c_0(j),c_s(j))=l$ for each $s<n_l$ and $j<k$ . This means that $f_{c_0(j)}|_l=f_{c_s(j)}|_l$ . Furthermore, since the elements of $\mathcal {C}$ are pairwise disjoint, then $\Xi _{c_s(j)}(l)=s$ . From the two previous facts, it is easy to see that for any $0<s<n_l$ and $j\in \omega $ , the order between $f_{c_0(j)}$ and $f_{c_s(j)}$ is decided by the value of both functions at l. Now, let us consider $0<i<n_{l}$ so that

$$ \begin{align*}C^{l-1}_{i}=\{\,\lVert c_0(j)\rVert _{l-1}\,: \,j<k \textit{ and }t(j)=\,<\,\}.\end{align*} $$

According to the second and third cases of the definitions of the functions, we have that $0<i=f_{c_i(j)}(l)$ whenever $t(j)=\,<$ and $0>-i=f_{c_i(j)}(l)$ whenever $t(j)=\,>$ . This implies $T(c_0,c_{i})=t,$ so we are done.

Theorem 5.8 (Under CA $^\Delta _2$ )

There is an increasing set.

Proof. Let $\mathcal {F}$ be a $2$ - $\Delta $ -capturing scheme of type $\langle m_k,n_{k+1},r_{k+1}\rangle _{k\in \omega }$ . For each $\alpha \in \omega _1$ , let $f_\alpha :\omega \longrightarrow \mathbb {Z}$ be given by

$$ \begin{align*}f_\alpha(k)=\Xi_\alpha(k).\end{align*} $$

By arguing in a similar way as we did in Theorem 5.7, we can conclude that $\mathcal {E}=\langle f_\alpha \rangle _{\alpha \in \omega _1}$ is an increasing set in $(\mathbb {Z}^\omega ,<_{lex})$ .

In [Reference Hrušák and Zindulka16], Hrušák and Zindulka investigated the (sometimes trivial) ideal $Mon(X)$ of all $\sigma $ -monotone subspaces of a given metric space $(X,d)$ . Particularly, they showed that every separable metric space of size less than the cardinal invariant $\mathfrak {m}_{\sigma -linked}$ is $\sigma $ -monotone. In other words, for all such spaces X, the ideal $Mon(X)$ is trivial. In Question 6.7 of that same paper, they asked:

$$ \begin{align*}\text{Is there a metric space of cardinality }\omega_1\text{ that is not }\sigma\text{-monotone?} \end{align*} $$

We will finish this section by proving that $\text {CA}^\Delta $ implies the existence of a metric space of cardinality $\omega _1$ which has no uncountable monotone subspaces. Therefore, it is consistent with arbitrarily large values of the continuum that the previous question has an affirmative answer.

By the previous remarks, it is easy to see that a space Y is not monotone if and only if it is not $\frac {1}{n}$ for each $0<n\in \omega $ . The following result is implicit in Lemma 4.4 and Proposition 4.5 of [Reference Nekvinda and Zindulka26].

Note that if m is a natural number for which there is a space $Z_n$ of size m which is not $\frac {1}{n}$ -monotone, then the same is true for each $m'\geqslant m$ .

Theorem 5.13 (Under $\text {CA}^\Delta $ )

There is a metric d over $\omega _1$ so that $(\omega _1,d)$ has no uncountable monotone subspaces.

Proof. Let $\tau =\langle m_k,n_{k+1},r_{k+1}\rangle _{k\in \omega } $ be a type so that for any $k>0$ , there is a metric $d_k$ over $n_k$ for which the following conditions holds:

$$ \begin{align*}\text{For each }0<k'\leqslant k, n_{k'}\text{ is not a } \frac{1}{k'}\text{-monotone subspace of }(n_k,d_k).\end{align*} $$

Such type exists due to the Lemma 5.12 and previous remarks. Furthermore, by Remark 5.9 we may assume that $diam(n_k)=1$ for each k.

Given $k\in \omega $ , define $s_k=\prod \limits _{i<k}\min (d_{i}(x,y)\,:\,x,y\in n_{i}\text { and } x\not =y)$ and let $\mathcal {F}$ be a $\Delta $ -capturing construction scheme of type $\tau $ . We define a metric over $\omega _1$ as follows:

$$ \begin{align*}d(\alpha,\beta)=\begin{cases}s_{\Delta(\alpha,\beta)}\cdot d_{\Delta(\alpha,\beta)}(\Xi_\alpha(\Delta(\alpha,\beta)),\Xi_\beta(\Delta(\alpha,\beta)))&\text{ if }\alpha\not=\beta\\ 0&\text{ if }\alpha=\beta. \end{cases} \end{align*} $$

In order to prove that $(\omega _1,d)$ satisfies the conclusion of the theorem, let $X\in [\omega _1]^{\omega _1}$ and $k\in \omega $ . We will prove that X is not $\frac {1}{k}$ -monotone. As $\mathcal {F}$ is $\Delta $ -capturing, we can find $D\in [X]^{n_k}$ which is $\Delta $ -captured at some level $l>k$ . By definition of d, note that $d(D(i),D(j))=s_l\cdot d_l(\Xi _{D(i)}(l),\Xi _{D(j)}(l))=s_l\cdot d_l(i,j)$ for any two $i,j<n_k$ . In other words, D is isometric to $n_k$ seen as a subspace of $(l,s_l\cdot d_l)$ . Therefore, D (and hence X) is not $\frac {1}{k}$ -monotone.

6 The basics on the forcing $\mathbb {P}(\mathcal {F})$

Let $\tau =\langle m_k,n_{k+1},r_{k+1}\rangle _{k\in \omega }$ be a (good) type. Recall that given $k\in \omega $ and Y a finite set of ordinals of size $m_k$ , there is a unique scheme over Y of type $\tau $ which we call $\mathcal {F}(Y)$ . There is also a unique construction scheme of type $\tau $ over $\omega $ . Such scheme, denoted by $\mathcal {F}(\omega )$ , can be defined as

$$ \begin{align*}\mathcal{F}( \omega):=\bigcup_{k\in \omega}\mathcal{F}(m_k).\end{align*} $$

In general, it can be proved that for an infinite $\lambda \leqslant \omega _1$ , there is a construction scheme of domain $\lambda $ if and only if $\lambda $ is limit.Footnote ⁴ Given a limit $\gamma \leqslant \lambda $ , we define $\mathcal {F}|_\gamma $ as $\{F\in \mathcal {F}\,:\,F\subseteq \gamma \}.$ This family is always a construction scheme with domain $\gamma $ . Furthermore, if $\mathcal {F}_\gamma $ is a scheme with domain $\gamma $ so that $\mathcal {F}_\gamma \subseteq \mathcal {F}$ , then $\mathcal {F}_\gamma =\mathcal {F}|_\gamma $ .

According to the previous paragraph we get that every scheme over $\omega _1$ is fully determined by a sequence of scheme $\langle \mathcal {F}_\gamma \rangle _{\gamma \in \text {Lim}}$ so that:

• • $\mathcal {F}_\gamma $ has domain $\gamma $ .

• • If $\gamma <\lambda $ , then $\mathcal {F}_\gamma \subseteq \mathcal {F}_\lambda $ .

• • If $\lambda $ is a limit of limit ordinals, then $\mathcal {F}_\lambda =\bigcup _{\gamma <\lambda }\mathcal {F}_\gamma .$

The main difficulty while building such a sequence is knowing how to extend a scheme $\mathcal {F}$ over a limit $\gamma $ to a new scheme over $\gamma +\omega $ . In order to deal with this situation, we use the forcing $\mathbb {P}(\mathcal {F})$ defined below. This forcing was first introduced by Todorčevíc in [Reference Todorcevic35] as a tool for building schemes in a recursive manner. Here, we follow the presentation from [Reference Chapital, González and Todorcevic9]. The reader can find the proofs for the results stated in this section in either of those papers. For the rest of this section, let us fix a construction scheme $\mathcal {F}$ of type $\tau $ over a limit ordinal $\gamma $ .

Definition 6.1 (The forcing $\mathbb {P}(\mathcal {F})$ )

We define $\mathbb {P}(\mathcal {F})$ as the forcing consisting of the empty set and of all $p\in \text {Fin}(\gamma +\omega )$ with the following properties:

1. (I) There is $k_p\in \omega $ such that $|p|=m_{k_p}$ .

2. (II) There is $F\in \mathcal {F}_{k_p}$ such that $p\cap \gamma \sqsubseteq F$ .

3. (III) $p\cap [\gamma ,\gamma +\omega )$ is an initial segment of $[\gamma ,\gamma +\omega )$ .

Whenever $p\cap \gamma \not =\emptyset $ (even if p is not a condition of $\mathbb {P}(\mathcal {F})$ ), we let $\alpha _p=\max (p\cap \gamma )$ . Additionally, for each $k\in \omega $ we let $\mathbb {P}_k(\mathcal {F})=\{\,p\in \mathbb {P}(\mathcal {F})\,:\,k_p=k\, \}.$ The order on $\mathbb {P}(\mathcal {F})$ is given by

$$ \begin{align*}p\leqslant q\text{ if and only if }q\in \mathcal{F}(p) \textit{or }q=\emptyset.\end{align*} $$

Definition 6.2 (The reduction operation)

If $p\in \text {Fin}(\gamma +\omega )$ and $\delta \leqslant \gamma $ , we define the reduction of p to $\delta $ as follows:

$$ \begin{align*}\mathrm{red}_\delta(p)=\begin{cases}(p\cap \delta)\cup[\max(p\cap \delta)+1,\max(p\cap \delta)+|p\backslash \delta|+1)& \textit{if }p\cap\delta\not=\emptyset\\ |p|& \textit{if }p\cap\delta=\emptyset. \end{cases} \end{align*} $$

Given a filter $\mathcal {G}$ over $\mathbb {P}(\mathcal {F})$ , we define $\mathcal {F}^{\mathcal {G}}$ as $\bigcup _{p\in \mathcal {G}}\mathcal {F}(p).$ Finally, $\mathcal {F}^{Gen}$ denotes the name for $\mathcal {F}^{\mathcal {G}}$ where G is a generic filter.

Definition 6.4 (The cut operation)

Let $F\in \text {Fin}(\gamma )$ and $\alpha \in \gamma $ . We define the cut of F at $\alpha $ as follows:

$$ \begin{align*}\mathrm{Cut}_\alpha(F)=(F\cap \alpha)\cup [\gamma,\gamma+|F\backslash\alpha|).\end{align*} $$

The reduction and cut operations are in some sense inverses of each other. That is, if $p\in \mathbb {P}(\mathcal {F})$ and $p\cap \gamma \not =\emptyset $ , then $\mathrm {Cut}_{\alpha _p+1}(\mathrm {red}_\gamma (p))=p$ . On the other hand, if $F\in \text {Fin}(\gamma )$ and $\alpha \in F$ then $\mathrm {red}_\gamma (\mathrm {Cut}_{\alpha +1}(F))=\mathrm {red}_{\alpha +1}(F)$ .

We now define the property $\mathrm {IH}_1$ . As it is stated in Proposition 6.8 below, this property allow us to extend a construction scheme with countable limit domain $\gamma $ to one whose domain is $\gamma +\omega $ .

One can prove that $\mathcal {F}(\omega )$ satisfies $\mathrm {IH}_1$ . Even more, if $\mathcal {F}$ is a scheme with $dom(\mathcal {F})=\lambda $ where $\lambda $ is a limit of limit ordinals, then $\mathcal {F}$ satisfies $\mathrm {IH}_1$ if and only if $\mathcal {F}|_\gamma $ satisfies $\mathrm {IH}_1$ for each limit $\gamma <\lambda .$

7 $\rho $ -capturing and the principle

A C-sequence over $\omega _1$ is a sequence $\mathcal {C}=\langle C_\delta \rangle _{\delta \in \mathrm {Lim}}$ of subsets of $\omega _1$ of order type $\omega $ with $\sup (C_\delta )=\delta $ for each $\delta \in \mathrm {Lim}$ . In the last 30 years, C-sequences have been proved to be a fundamental tool for solving major problems in set theory and topology. The reader can consult [Reference Todorcevic34] for more information on C-sequences both over $\omega _1$ and over higher cardinals. The goal of this section is to prove that -principle implies $CA^\rho $ . For that, the only thing we need to know is that if -principle holds, then there is a C-sequence which is moreover a -sequence.

In what follows, we will fix a C-sequence $\mathcal {C}=\langle C_\delta \rangle _{\delta \in \text {Lim}}$ and a type $\tau =\langle m_k,n_{k+1},r_{k+1}\rangle _{k\in \omega }$ . Note that apart from Theorem 7.11, the analysis presented here works even if $\mathcal {C}$ is not assumed to be sequence.

Let us define $\mathrm {Lim}_0:=\mathrm {Lim}$ and $\mathrm {Lim}_{n+1}:=\{\delta \in \mathrm {Lim}\,:\,C_\delta \subseteq \mathrm {Lim}_n\}$ for each $n\in \omega $ . Given $\alpha \in \text {Lim}_0$ , put $C^0_\alpha :=C_\alpha $ . Finally, for each $n\in \omega $ and $\delta \in \text {Lim}_{n+1}$ , let