COFINAL TYPES BELOW $\aleph _\omega $

Abstract

It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph _n$ is at least $c_{n+2}$, the $(n+2)$-Catalan number. Moreover, the class $\mathcal D_{\aleph _n}$ of directed sets of cardinality $\leq \aleph _n$ contains an isomorphic copy of the poset of Dyck $(n+2)$-paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

1 Introduction

Motivated by problems in general topology, Birkhoff [1], Tukey [15], and Day [2] studied some natural classes of directed sets. Later, Schmidt [9] and Isbell [4, 5] investigated uncountable directed sets under the Tukey order $<_T$ . In [12], Todorčević showed that under ${\mathsf {PFA}}$ there are only five cofinal types in the class $\mathcal D_{\aleph _1}$ of all cofinal types of size $\leq \aleph _1$ under the Tukey order, namely, $\{1,\omega ,\omega _1,\omega \times \omega _1,[\omega _1]^{<\omega }\}$ . In the other direction, Todorčević showed that under $\operatorname {\mathrm {CH}}$ there are $2^{\mathfrak c}$ many non-equivalent cofinal types in this class. Later in [14] this was extended to all transitive relations on $\omega _1$ . Recently, Kuzeljević and Todorčević [6] initiated the study of the class $\mathcal D_{\aleph _2}$ . They showed in $\operatorname {\mathrm {ZFC}}$ that this class contains at least fourteen different cofinal types which can be constructed from two basic types of directed sets and their products: $(\kappa ,\in )$ and $([\kappa ]^{<\theta },\subseteq )$ , where $\kappa \in \{1,\omega ,\omega _1,\omega _2\}$ and $\theta \in \{\omega ,\omega _1\}$ .

In this paper, we extend the work of Todorčević and his collaborators and uncover a connection between the classes of the $\mathcal D_{\aleph _n}$ ’s and the Catalan numbers. Denote $V_k:=\{1, \omega _k, [\omega _k]^{<\omega _m} \mid 0\leq m<k\}$ , $\mathcal F_n:=\bigcup _{k\leq n} V_k$ and finally let $\mathcal S_n$ be the set of all finite products of elements of $\mathcal F_n$ . Recall (see Section 3) that the n-Catalan number is equal to the cardinality of the set of all Dyck n-paths. The set $\mathcal K_n$ of all Dyck n-paths admits a natural ordering $\vartriangleleft $ , and the connection we uncover is as follows.

Theorem A. The posets and $(\mathcal K_{n+2},\vartriangleleft )$ are isomorphic. In particular, the class $\mathcal D_{\aleph _n}$ has size at least the $(n+2)$ -Catalan number.

A natural question which arises is whether an interval determined by two successive elements of forms an empty interval in $(\mathcal D_{ \aleph _n},<_T)$ . In [6], the authors showed that there are two intervals of $\mathcal S_2$ that are indeed empty in $\mathcal D_{\aleph _2}$ , and they also showed that consistently, under $\operatorname {\mathrm {GCH}}$ and the existence of a non-reflecting stationary subset of $E^{\omega _2}_\omega $ , two intervals of $\mathcal S_2$ that are nonempty in $\mathcal D_{\aleph _2}$ .

In this paper, we prove:

Theorem B. Assuming $\operatorname {\mathrm {GCH}}$ , for every positive integer n, all intervals of $\mathcal S_n$ that form an empty interval in $\mathcal D_{\aleph _n}$ are identified, and counterexamples are constructed to the other cases.

1.1 Organization of this paper

In Section 2 we analyze the Tukey order of directed sets using characteristics of the ideal of bounded subsets.

In Section 3 we consider the poset and show it is isomorphic to the poset of good $(n+2)$ -paths (Dyck paths) with the natural order. As a corollary we get that the cardinality of $\mathcal D_{ \aleph _n}$ is greater than or equal to the Catalan number $c_{n+2}$ . Furthermore, we address the basic question of whether a specific interval in the poset is empty, i.e., considering an element C and a successor of it E, is there a directed set $D\in \mathcal D_{\aleph _n}$ such that $C<_T D <_T E$ ? We answer this question in Theorem 3.5 using results from the next two sections.

In Section 4 we present sufficient conditions on an interval of the poset which enable us to prove there is no directed set inside.

In Section 5 we present cardinal arithmetic assumptions, enough to construct on specific intervals of the poset a directed set inside.

In Section 6 we finish with a remark about future research.

In the Appendix diagrams of the posets and are presented.

1.2 Notation

For a set of ordinals C, we write $\operatorname {\mathrm {acc}}(C):=\{\alpha <\sup (C) \mid \sup (C\cap \alpha )=\alpha>0\}$ . For $\alpha <\gamma $ where $\alpha $ is a regular cardinal, denote $E^\gamma _\alpha :=\{ \beta <\gamma \mid \operatorname {\mathrm {cf}}(\beta )=\alpha \}$ . The set of all infinite (resp. infinite and regular) cardinals below $\kappa $ is denoted by $\operatorname {\mathrm {Card}}(\kappa )$ (resp. $\operatorname {\mathrm {Reg}}(\kappa )$ ). For a cardinal $\kappa $ we denote by $\kappa ^+$ the successor cardinal of $\kappa $ , and by $\kappa ^{+n}$ the nth-successor cardinal. For a function $f:X\rightarrow Y$ and a set $A\subseteq X$ , we denote $f"A:=\{f(x)\mid x\in A\}$ . For a set A and a cardinal $\theta $ , we write $[A]^{\theta }:=\{X\subseteq A \mid |X|=\theta \}$ and define $[A]^{\leq \theta }$ and $[A]^{<\theta }$ similarly. For a sequence of sets $\langle A_i\mid i\in A \rangle $ , let $\prod _{i\in I} D_i:=\{f:I\rightarrow \bigcup _{i\in I} D_i \mid \forall i\in I[f(i)\in D_i] \}$ .

1.3 Preliminaries

A partial ordered set $(D,\leq _D)$ is directed iff for every $x,y\in D$ there is $z\in D$ such that $x\leq _D z$ and $y\leq _D z$ . We say that a subset X of a directed set D is bounded if there is some $d\in D$ such that $x\leq _D d$ for each $x\in X$ . Otherwise, X is unbounded in D. We say that a subset X of a directed D is cofinal if for every $d\in D$ there exists some $x\in X$ such that $d\leq _D x$ . Let $\operatorname {\mathrm {cf}}(D)$ denote the minimal cardinality of a cofinal subset of D. If D and E are two directed sets, we say that $f:D\rightarrow E$ is a Tukey function if $f"X:=\{f(x)\mid x\in X\}$ is unbounded in E whenever X is unbounded in D. If such a Tukey function exists we say that D is Tukey reducible to E, and write $D\leq _T E$ . If $D\leq _T E$ and $E\not \leq _T D$ , we write $D<_T E$ . A function $g:E\rightarrow D$ is called a convergent/cofinal map from E to D if for every $d\in D$ there is an $e_d\in E$ such that for every $c\geq e_d$ we have $g(c)\geq d$ . There is a convergent map $g:E\rightarrow D$ iff $D\leq _T E$ . Note that for a convergent map $g:E\rightarrow D$ and a cofinal subset $Y\subseteq E$ , the set $g"Y$ is cofinal in D. We say that two directed sets D and E are cofinally/Tukey equivalent and write $D\equiv _T E$ iff $D\leq _T E$ and $D\geq _T E$ . Formally, a cofinal type is an equivalence class under the Tukey order, we abuse the notation and call every representative of the class a cofinal type. Notice that a directed set D is cofinally equivalent to any cofinal subset of D. In [15], Tukey proved that $D\equiv _T E$ iff there is a directed set $(X, \leq _X)$ such that both D and E are isomorphic to a cofinal subset of X. We denote by $\mathcal D_\kappa $ the set of all cofinal types of directed sets of cofinality $\leq \kappa $ .

Consider a sequence of directed sets $\langle D_i \mid i\in I \rangle $ , we define the directed set which is the product of them $(\prod _{i\in I} D_i,\leq )$ ordered by everywhere-dominance, i.e., for two elements $d, e \in \prod _{i\in I} D_i$ we let $ d \leq e$ if and only if $ d(i)\leq _{D_i} e(i)$ for each $i\in I$ . For $X\subseteq \prod _{i\in I} D_i$ , let $\pi _{D_i}$ be the projection to the i-coordinate. A simple observation [12, Proposition 2] is that if n is finite, then $D_1\times \dots \times D_n$ is the least upper bound of $D_1,\dots ,D_n$ in the Tukey order. Similarly, we define a $\theta $ -support product $\prod ^{\leq \theta }_{i\in I}D_i$ ; for each $i\in I$ , we fix some element $0_{D_i}\in D_i$ (usually minimal). Every element $v\in \prod ^{\leq \theta }_{i\in I}D_i$ is such that $|\operatorname {\mathrm {supp}}(v)|\leq \theta $ , where $\operatorname {\mathrm {supp}}(v):=\{ i\in I \mid v(i)\not = 0_{D_i}\}$ . The order is coordinate wise.

2 Characteristics of directed sets

We commence this section with the following two lemmas which will be used throughout the paper.

Lemma 2.1 (Pouzet [7]).

Suppose D is a directed set such that $\operatorname {\mathrm {cf}}(D)=\kappa $ is infinite, then there exists a cofinal directed set $P\subseteq D$ of size $\kappa $ such that every subset of size $\kappa $ of P is unbounded

Proof Let $X\subseteq D$ be a cofinal subset of cardinality $\kappa $ and let $\{x_\alpha \mid \alpha <\kappa \}$ be an enumeration of X. Let $P:=\{ x_\alpha \mid \alpha <\kappa \text { and for all } \beta <\alpha [x_\alpha \not <_D x_\beta ] \}$ . We claim that P is cofinal. In order to prove this, fix $d\in D$ . As X is cofinal in D, fix a minimal $\alpha <\kappa $ such that $d<_D x_\alpha $ . If $x_\alpha \in P$ , then we are done. If not, then fix some $\beta <\alpha $ minimal such that $x_\alpha <_D x_\beta $ . We claim that $x_\beta \in P$ , i.e., there is no $\gamma <\beta $ such that $x_\beta <_D x_\gamma $ . Suppose there is some $\gamma <\beta $ such that $x_\beta <_D x_\gamma $ , then $x_\alpha <_D x_\gamma $ , which is a contradiction to the minimality of $\beta $ . Note that $d<_D x_\beta \in P$ as sought. As P is cofinal in D, $\operatorname {\mathrm {cf}}(D)=\kappa $ , $P\subseteq X$ and $|X|=\kappa $ , we get that $|P|=\kappa $ .

Finally, let us show that every subset of size $\kappa $ of P is unbounded. Suppose on the contrary that $X\subseteq P$ is a bounded subset of P of size $\kappa $ . Fix some $x_\beta \in P$ above X and $\beta <\alpha <\kappa $ such that $x_\alpha \in X$ , but this is an absurd as $x_\alpha <_D x_\beta $ and $x_\alpha \in P$ .

Fact 2.2 (Kuzeljević–Todorčević [6, Lemma 2.3]).

Let $\lambda \geq \omega $ be a regular cardinal and $n<\omega $ be positive. The directed set $[\lambda ^{+n}]^{\leq \lambda }$ contains a cofinal subset $\mathfrak D_{[\lambda ^{+n}]^{\leq \lambda }}$ of size $\lambda ^{+n}$ with the property that every subset of $\mathfrak D_{[\lambda ^{+n}]^{\leq \lambda }}$ of size $>\lambda $ is unbounded in $[\lambda ^{+n}]^{\leq \lambda }$ . In particular, $[\lambda ^{+n}]^{\leq \lambda }$ belongs to $\mathcal D_{\lambda ^{+n}}$ , i.e. $\operatorname {\mathrm {cf}}([\lambda ^{+n}]^{\leq \lambda }) \leq \lambda ^{+n}$ .

Recall that any directed set is Tukey equivalent to any of its cofinal subsets, hence $\mathfrak D_{[\lambda ^{+n}]^{\leq \lambda }} \equiv _T [\lambda ^{+n}]^{\leq \lambda }$ .

As part of our analysis of the class $\mathcal D_{\aleph _n}$ , we would like to find certain traits of directed sets which distinguish them from one another in the Tukey order. This was done previously, in [4, 9, 14]. We use that the language of cardinal functions of ideals.

Definition 2.3. For a set D and an ideal $\mathcal I$ over D, consider the following cardinal characteristics of $\mathcal I$ :

• • $\operatorname {\mathrm {add}}(\mathcal I):=\min \{\kappa \mid \mathcal A \subseteq I,~ |\mathcal A|=\kappa , ~\bigcup \mathcal A\notin \mathcal I\}$ ;

• • $ \operatorname {\mathrm {non}}(\mathcal I):=\min \{|X|\mid X\subseteq D,~ X\notin \mathcal I \}$ ;

• • $ \operatorname {\mathrm {out}}(\mathcal I):=\min \{ \theta \leq |D|^+\mid \mathcal I\cap [D]^{\theta }=\emptyset \}$ ;

• • $\operatorname {\mathrm {in}}(\mathcal I, \kappa ) = \{ \theta \leq \kappa \mid \forall X\in [D]^\kappa ~\exists Y\in [X]^\theta \cap \mathcal I \}$ .

Notice that $\operatorname {\mathrm {add}}(\mathcal I)\leq \operatorname {\mathrm {non}}(\mathcal I) \leq \operatorname {\mathrm {out}}(\mathcal I)$ .

Definition 2.4. For a directed set D, denote by $\mathcal I_{\operatorname {\mathrm {bd}}}(D) $ the ideal of bounded subsets of D.

Proposition 2.5. Let D be a directed set. Then $:$

1. (1) $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))$ is the minimal size of an unbounded subset of D, so every subset of size less than $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))$ is bounded.

2. (2) If $\theta <\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))$ , then there exists in D some bounded subset of size $\theta $ .

3. (3) If $\theta \geq \operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))$ , then every subset X of size $\theta $ is unbounded in D.

4. (4) If $\theta \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D),\kappa )$ , then for every $X\in [D]^\kappa $ there exists some $B\in [X]^{\theta }$ bounded.

5. (5) For every $\theta <\operatorname {\mathrm {add}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))$ and a family $\mathcal A$ of size $\theta $ of bounded subsets of D, the subset $\bigcup \mathcal A$ is also bounded in D.

Let us consider another intuitive feature of a directed set, containing information about the cardinality of hereditary unbounded subsets, this was considered previously by Isbell [4].

Definition 2.6 (Hereditary unbounded sets).

For a directed set D, set

$$ \begin{align*}\operatorname{\mathrm{hu}}(D):=\{ \kappa\in \operatorname{\mathrm{Card}}(|D|^+)\mid \exists X\in [D]^{\kappa}[\forall Y\in [X]^\kappa\text{ is unbounded}] \}.\end{align*} $$

Proposition 2.7. Let D be a directed set. Then $:$

• • If $\operatorname {\mathrm {cf}}(D)$ is an infinite cardinal, then $\operatorname {\mathrm {cf}}(D)\in \operatorname {\mathrm {hu}}(D)$ .

• • If $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))\leq \kappa \leq |D|$ , then $ \kappa \in \operatorname {\mathrm {hu}}(D)$ .

• • For an infinite cardinal $\kappa $ we have that $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(\kappa ))=\operatorname {\mathrm {cf}}(\kappa )$ , $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))=\kappa $ and $\operatorname {\mathrm {hu}}(\kappa )=\{\lambda \in \operatorname {\mathrm {Card}}(\kappa ^+) \mid \lambda =\operatorname {\mathrm {cf}}(\kappa )\}$ .

• • If $\kappa =\operatorname {\mathrm {cf}}(D)=\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))$ , then $D\equiv _T\kappa $ .

• • For two infinite cardinals $\kappa>\theta $ we have that $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}([\kappa ]^{<\theta }))=\operatorname {\mathrm {cf}}(\theta )$ .

• • For a regular cardinal $\kappa $ and a positive $n<\omega $ , $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(\mathfrak D_{[\kappa ^{+n}]^{\leq \kappa }}))>\kappa $ and $\operatorname {\mathrm {hu}}(\mathfrak D_{[\kappa ^{+n}]^{\leq \kappa }}) = \{ \kappa ^{+(m+1)}\mid m< n\}$ .

• • If $\kappa =\operatorname {\mathrm {cf}}(D)$ is regular, $\theta =\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))=\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))$ and $\theta ^{+n}=\kappa $ for some $n<\omega $ , then $D\equiv _T [\kappa ]^{<\theta }$ .

In the rest of this section we consider various scenarios in which the traits of a certain directed set give us information about its position in the poset $(\mathcal D_{\kappa }, <_T )$ .

Lemma 2.8. Suppose D is a directed set, $\kappa $ is an infinite regular cardinal and $X\subseteq D$ is an unbounded subset of size $\kappa $ such that every subset of X of size $<\kappa $ is bounded. Then $\kappa \in \operatorname {\mathrm {hu}}(D)$ .

Proof Enumerate $X:=\{x_\alpha \mid \alpha <\kappa \}$ , by the assumption, for every $\alpha <\kappa $ we may fix some $z_\alpha \in D$ above the bounded initial segment $\{x_\beta \mid \beta <\alpha \}$ . We show that $Z:=\{z_\alpha \mid \alpha <\kappa \}$ , witnesses $\kappa \in \operatorname {\mathrm {hu}}(D)$ . First, let us show that $|Z|=\kappa $ . Suppose on the contrary that $Z:=\{z_\alpha \mid \alpha <\kappa \}$ is of cardinality $< \kappa $ . Then for some $\alpha <\kappa $ , the element $z_\alpha $ is above the subset X, hence X is bounded which is absurd. Now, let us prove that Z is hereditarily unbounded. We claim that every subset of Z of cardinality $\kappa $ is also unbounded. Suppose not, let us fix some $W\in [Z]^\kappa $ bounded by some $d\in D$ , but then d is above X contradicting the fact that X is unbounded.

Lemma 2.9. Suppose D is a directed set and $\kappa $ is an infinite cardinal in $\operatorname {\mathrm {hu}}(D)$ , then $\kappa \leq _TD$ .

Proof Fix $X\subseteq D$ of cardinality $\kappa $ such that every subset of X of size $\kappa $ is unbounded and a one-to-one function $f:\kappa \rightarrow X$ , notice that f is a Tukey function from $\kappa $ to D as sought.

Corollary 2.10. Suppose D is directed set, $\kappa $ is regular and $X\subseteq D$ is an unbounded subset of size $\kappa $ such that every subset of X of size $< \kappa $ is bounded, then $\kappa \leq _T D$ .

The reader may check the following:

• • For any two infinite cardinals $\lambda $ and $\kappa $ of the same cofinality, we have $\lambda \equiv _T \kappa $ .

• • For an infinite regular cardinal $\kappa $ , we have $\kappa \equiv _T [\kappa ]^{<\kappa }$ .

• • $\operatorname {\mathrm {hu}}(\prod ^{<\omega }_{n<\omega }\omega _{n+1}) = \{\omega _n \mid n<\omega \}$ .

Lemma 2.11. Suppose D and E are two directed sets such that for some $\theta \in \operatorname {\mathrm {hu}}(D)$ regular we have $\theta>\operatorname {\mathrm {cf}}(E)$ , then $D\not \leq _T E$ .

Proof By passing to a cofinal subset, we may assume that $|E|=\operatorname {\mathrm {cf}}(E)$ . Fix $\theta \in \operatorname {\mathrm {hu}}(D)$ regular such that $\operatorname {\mathrm {cf}}(E)<\theta $ and $X\in [D]^\theta $ witnessing $\theta \in \operatorname {\mathrm {hu}}(D)$ , i.e., every subset of X of size $\theta $ is unbounded. Suppose on the contrary that there exists a Tukey function $f:D\rightarrow E$ . By the pigeonhole principle, there exists some $Z\in [X]^{\theta }$ and $e\in E$ such that $f"Z=\{e\}$ . As f is Tukey and the subset $Z\subseteq X$ is unbounded, $f"Z$ is unbounded in E which is absurd.

Notice that for every directed set D, if $\operatorname {\mathrm {cf}}(D)>1$ , then $\operatorname {\mathrm {cf}}(D)$ is an infinite cardinal.

As a corollary from the previous lemma, $\lambda \not \leq _T \kappa $ for any two regular cardinals $\lambda>\kappa $ where $\lambda $ is infinite. Furthermore, the reader can check that $\lambda \not \leq _T\kappa $ , whenever $\lambda <\kappa $ are infinite regular cardinals.

Lemma 2.12. Suppose C and D are directed sets such that $C\leq _T D$ , then $\operatorname {\mathrm {cf}}(C)\leq \operatorname {\mathrm {cf}}(D)$ .

Proof Suppose $|D|=\operatorname {\mathrm {cf}}(D)$ and let $f:C\rightarrow D$ be a Tukey function. As f is Tukey, for every $d\in D$ the set $\{x\in C\mid f(x)=d\}$ is bounded in C by some $c_d\in C$ . Note that for every $x\in C$ , we have $x\leq _C c_{f(x)}$ , hence the set $\{c_d \mid d\in D\}$ is cofinal in C. So $\operatorname {\mathrm {cf}}(C)\leq |D|=\operatorname {\mathrm {cf}}(D)$ as sought.

Lemma 2.13. Let $\kappa $ and $\theta $ be two cardinals such that $\theta <\kappa =\operatorname {\mathrm {cf}}(\kappa )$ .

Suppose D is a directed set such that $\operatorname {\mathrm {cf}}(D)\leq \kappa $ and $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))\geq \theta $ , then $D\leq _T [\kappa ]^{<\theta }$ . Furthermore, if $\theta \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D),\kappa )$ , then $D<_T [\kappa ]^{<\theta }$ .

Proof First, we show that there exists a Tukey function $f:D \rightarrow [\kappa ]^{<\theta }$ . Let us fix a cofinal subset $X\subseteq D$ of cardinality $\leq \kappa $ such that every subset of X of cardinality $< \theta $ is bounded. As $|X|\leq \kappa $ we may fix an injection $f:X\rightarrow [\kappa ]^{1}$ , we will show f is a Tukey function. Let $Y\subseteq X$ be a subset unbounded in D, this implies $|Y|\geq \theta $ . As f is an injection, the set $\bigcup f"Y$ is of cardinality $\geq \theta $ . Note that every subset of $[\kappa ]^{<\theta }$ whose union is of cardinality $\geq \theta $ is unbounded in $[\kappa ]^{<\theta }$ , hence $f"Y$ is an unbounded subset in $[\kappa ]^{<\theta }$ as sought.

Assume $\theta \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D),\kappa )$ , we are left to show that $[\kappa ]^{<\theta }\not \leq _T D$ . Suppose on the contrary that $g:[\kappa ]^{<\theta }\rightarrow D$ is a Tukey function. We split to two cases:

$\blacktriangleright $ Suppose $| g" [\kappa ]^1| < \kappa $ . As $\kappa $ is regular, by the pigeonhole principle there exists a set $X\subseteq [\kappa ]^1$ of cardinality $\kappa $ , and $d\in D$ such that $g(x)=d$ for each $x\in X$ . Notice $g"X$ is a bounded subset of D. As $X\subseteq [\kappa ]^1$ is of cardinality $\kappa $ and $\kappa>\theta $ , it is unbounded in $[\kappa ]^{<\theta }$ . Since g is a Tukey function, we get that $g"X$ is unbounded which is absurd.

$\blacktriangleright $ Suppose $|g"[\kappa ]^1| = \kappa $ . Let $X:=g"[\kappa ]^1$ , by our assumption on D, there exists a bounded subset $B\in [X]^{\theta }$ . Since B is of size $\theta $ , we get that $(g^{-1} [B])\cap [\kappa ]^1$ is of cardinality $\geq \theta $ , hence unbounded in $[\kappa ]^{<\theta }$ , which is absurd to the assumption g is Tukey.

Remark 2.14. For every two directed sets, D and E, if $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D) )<\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E) )$ , then $D\not \leq _T E$ . For example, $\theta \not \leq _T [\kappa ]^{\leq \theta }$ .

Lemma 2.15. Let $\kappa $ be a regular infinite cardinal. Suppose D and E are two directed sets such that $|D|\geq \kappa $ and $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))\in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E),\kappa )$ , then $D\not \leq _T E$ .

Proof Let $\theta :=\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))$ . By the definition of $\operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E),\kappa )$ , as $\theta \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E),\kappa )$ , we know that $\theta \leq \kappa $ . Notice that every subset of D of size $\geq \theta $ is unbounded in D and every subset of size $\kappa $ of E contains a bounded subset in E of size $\theta $ .

Suppose on the contrary that there exists a Tukey function $f:D\rightarrow E$ . We split to two cases:

$\blacktriangleright $ Suppose $|f"D|<\kappa $ , then by the pigeonhole principle there exists some $X\in [D]^\kappa $ and $e\in E$ such that $f"X=\{e\}$ . As $|X|=\kappa \geq \theta $ , we know that X is unbounded in D, but $f"X$ is bounded in E which is absurd as f is a Tukey function.

$\blacktriangleright $ Suppose $|f"D|\geq \kappa $ , by the assumption there exists a subset $Y\in [f"D]^\theta $ which is bounded in E. Notice that $X:=f^{-1}Y$ is a subset of D of size $\geq \theta $ , hence unbounded in D. So X is an unbounded subset of D such that $f"X=Y$ is bounded in E, contradicting the fact that f is a Tukey function.

Lemma 2.16. Suppose $\kappa $ is a regular uncountable cardinal, C and $\langle D_m \mid m<n\rangle $ are directed sets such that $|C|<\kappa \leq \operatorname {\mathrm {cf}}(D_m)$ and $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D_m))>\theta $ for every $m<n$ . Then $\theta \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}} (C\times \prod _{m<n}{D_m}),\kappa ) $ .

Proof Suppose $X\subseteq C\times \prod _{m<n}{D_m}$ is of size $\kappa $ , we show that X contains a bounded subset of size $\theta $ . As $|C|<\kappa $ , by the pigeonhole principle we can fix some $Y\in [X]^\kappa $ and $c\in C$ such that $\pi _C " Y=\{c\}$ . Suppose on the contrary that some subset $Z\subseteq Y$ of size $\theta $ is unbounded, it must be that for some $m<n$ the set $\pi _{D_m}"Z$ is unbounded in $D_m$ , but this is absurd as $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D_m))>\theta $ and $|\pi _{D_m}"Z|\leq \theta $ .

Lemma 2.17. Suppose $C,D$ and E are directed sets such that $:$

• • for every partition $D=\bigcup _{\gamma <\kappa } D_\gamma $ , there exists an ordinal $\gamma <\kappa $ , and an unbounded $X\subseteq D_\gamma $ of size $\kappa $ ;

• • $|C|\leq \kappa $ ;

• • $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))> \kappa $ .

Then $D\not \leq _T C\times E$ .

Proof Suppose on the contrary, that there exists a Tukey function $h:D \rightarrow C \times E$ . For $c\in C$ , let $D_c:=\{x\in D \mid \exists e\in E [h(x)=(c,e)]\}$ . Since h is a function, $D:=\bigcup _{c\in C} D_c$ is a partition to $\leq \kappa $ many sets. By the assumption, there exists $c\in C$ and an unbounded subset $X\subseteq D_c$ of cardinality $\kappa $ . Enumerate $X=\{x_\xi \mid \xi <\kappa \}$ and let $e_\xi \in E$ be such that $h(x_\xi ) =(c,e_\xi )$ , for each $\xi <\kappa $ . As $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))> \kappa $ , there exists some upper bound $e\in E$ to the set $\{e_\xi \mid \xi <\kappa \}$ . Since X is unbounded and h is Tukey, $h"X=\{(c,e_\xi )\mid \xi <\kappa \}$ must be unbounded, which is absurd as $(c,e)$ is bounding it.

Note that the lemma is also true when the partition of D is of size less than $\kappa $ .

3 The Catalan structure

The sequence of Catalan numbers $\langle c_n \mid n<\omega \rangle =\langle 1,1,2,5,14,42,\dots \rangle $ is an ubiquitous sequence of integers with many characterizations, for a comprehensive review of the subject, we refer the reader to Stanley’s book [11]. One of the many representations of $c_n$ , is the number of good n-paths (Dyck paths), where a good n-path is a monotonic lattice path along the edges of a grid with $n\times n$ square cells, which do not pass above the diagonal. A monotonic path is one which starts in the lower left corner, finishes in the upper right corner, and consists entirely of edges pointing rightwards or upwards. An equivalent representation of a good n-path, which we will consider from now on, is a vector $\vec p$ of the columns’ heights of the path (ignoring the first trivial column), i.e., a vector $\vec p=\langle p_0,\dots ,p_{n-2}\rangle $ of length $n-1$ of $\leq $ -increasing numbers satisfying $0\leq p_k\leq k+1$ , for every $0\leq k\leq n-2$ . We consider the poset $(\mathcal K_n,\vartriangleleft )$ where $\mathcal K_n$ is the set of all good n-paths and the relation $\vartriangleleft $ is defined such that $\vec a \vartriangleleft \vec b$ if and only if the two paths are distinct and for every k with $0\leq k\leq n-2$ we have $b_k\leq a_k$ , in other words, the path $\vec b$ is below the path $\vec a$ (allowing overlaps). Notice that for two distinct good n-paths $\vec a$ and $\vec b$ , either $\vec a \not \vartriangleleft \vec b$ or $\vec b \not \vartriangleleft \vec a$ . A good n-path $\vec b$ is an immediate successor of a good n-path $\vec a$ if $\vec a\vartriangleleft \vec b$ and $\vec a-\vec b$ is a vector with value $0$ at all coordinates except one of them which gets the value $1$ .

Figure 1 The good $4$ -path $\langle 1,1,3\rangle $ .

Suppose $\vec a$ and $\vec b$ are two good n-paths where $\vec b$ is an immediate successor of $\vec a$ . Let $i\leq n-2$ be the unique coordinate on which $\vec a$ and $\vec b$ are different and $a_i$ be the value of $\vec a$ on this coordinate, i.e., $a_i=b_i+1 $ . We say that the pair $(\vec a, \vec b)$ is on the k-diagonal if and only if $i+1-a_i=k$ and $\vec b$ is an immediate successor of $\vec a$ (Figure 1).

In this section we show the connection between the Catalan numbers and cofinal types. Let us fix $n<\omega $ . Recall that for every $k<\omega $ , we set $V_k:=\{1, \omega _k, [\omega _k]^{<\omega _m} \mid 0\leq m<k\}$ , $\mathcal F_n:=\bigcup _{k\leq n} V_k$ and let $\mathcal S_n$ be the set of all finite products of elements in $\mathcal F_n$ . Our goal is to construct a coding which gives rise to an order-isomorphism between and $(\mathcal K_{n+2},\vartriangleleft )$ .

To do that, we first consider a “canonical form” of directed sets in $\mathcal S_n$ . By Lemma 2.13 the following hold:

1. (a) For all $0\leq l<m<k<\omega $ we have $1<_T \omega _k<_T [\omega _k]^{<\omega _m} <_T [\omega _k]^{<\omega _l} $ .

2. (b) For all $0\leq l\leq t<m\leq k<\omega $ with $(l,k)\neq (t,m)$ we have $[\omega _m]^{<\omega _t} <_T [\omega _k]^{<\omega _l}$ and $\omega _m <_T [\omega _k]^{<\omega _l}$ .

Notice that (a) implies $(V_k,<_T)$ is linearly ordered. A basic fact is that for two directed sets C and D such that $C\leq _T D$ , we have $C\times D \equiv _T D$ . Hence, for every $D\in \mathcal S_n$ we can find a sequence of elements $\langle D^k \mid k\leq n \rangle $ , where $D^k\in V_k$ for every $k\leq n$ , such that $D\equiv _T \prod _{k\leq n}D^k$ . As we are analyzing the class $\mathcal D_{\aleph _n}$ under the Tukey relation $<_T$ , two directed sets which are of the same $\equiv _T$ -equivalence class are indistinguishable, so from now on we consider only elements of this form in $\mathcal S_n$ .

We define a function $\mathfrak F:\mathcal S_n \rightarrow \mathcal S_n$ as follows: Fix $D\in \mathcal S_n$ where $D=\prod _{k\leq n} D^k$ . Next, we construct a sequence $\langle D_k \mid k\leq n \rangle $ by reverse recursion on $k\leq n$ . At the top case, set $D_n:=D^n$ . Next, for $0\leq k< n$ . If by (b), we get that $D^{k} <_T D^{m}$ for some $k<m\leq n$ , then set $D_{k}:=1$ . Else, let $D_{k}:=D^{k}$ . Finally, let $\mathfrak F(D) := \prod _{k\leq n} D_{k}$ . Notice that we constructed $\mathfrak F(D)$ such that $\mathfrak F(D) \equiv _T D$ . We define $\mathcal T_n:=\operatorname {\mathrm {Im}}(\mathfrak F)$ .

The coding. We encode each product $D\in \mathcal T_n$ by an $(n+2)$ -good path $\vec v_D:=\langle v_0,\dots , v_{n}\rangle $ . Recall that $D:=\prod _{k\leq n} D_k$ , where $D_k\in V_k$ for every $k\leq n$ . We define by reverse recursion on $0\leq k \leq n$ , the elements of the vector $\vec v_D$ such that $v_k\leq k+1$ as follows: Suppose one of the elements of $\langle [\omega _k]^{<\omega },\dots , [\omega _k]^{<\omega _{k-1}} ,\omega _k \rangle $ is equal to $D_k$ , then let $v_{k}$ be its coordinate (starting from $0$ ). Suppose this is not the case, then if $k=n$ , we let $v_{k}:=n+1$ else $v_{k}:=\min \{ v_{k+1}, k+1\}$ .

Notice that by (b), if $0\leq i<j\leq n$ , then $v_i \leq v_j$ . Hence, every element $D\in \mathcal T_n$ is encoded as a good $(n+2)$ -path.

To see that the coding is one-to-one, suppose $C,D\in \mathcal T_n$ are distinct. Let $k:=\max \{ i\leq n \mid C_i\neq D_i\}$ . We split to two cases:

$\blacktriangleright $ Suppose both $C_k$ and $D_k$ are not equal to $1$ , then clearly the column height of $\vec v_C$ and $\vec v_D$ are different at coordinate $k+1$ .

$\blacktriangleright $ Suppose one of them is equal to $1$ , say $C_k$ , then $D_k\neq 1$ . Let $\vec v_C:=\langle v^C_0,\dots v^C_n\rangle $ and $\vec v_D:=\langle v^D_0,\dots v^D_n\rangle $ . Suppose $k=n$ , then clearly $v^D_n < v^C_n$ . Suppose $k<n$ , then $v^D_i = v^C_i$ for $k<i\leq n$ . By the coding, $v^D_k<k+1$ and by (b) $v^D_k<v^D_{k+1}=v^C_{k+1}$ , but $v^C_k:=\min \{ k+1, v^C_{k+1}\}$ . Hence $v^D_k < v^C_k$ as sought.

To see that the coding is onto, let us fix a good $(n+2)$ -path $\vec v:=\langle v_0, \dots , v_n \rangle $ . We construct $\langle D_k \mid k\leq n \rangle $ by reverse recursion on $k\leq n$ . At the top case, set $D_n$ to be the $v_n$ element of the vector $\langle [\omega _n]^{<\omega },\dots , [\omega _n]^{<\omega _{n-1}} ,\omega _n, 1\rangle $ . For $k<n$ , if $v_k=v_{k+1}$ , let $D_{k}:=1$ . Else, let $D_k$ be the kth element of the vector $\langle [\omega _k]^{<\omega },\dots , [\omega _k]^{<\omega _{k-1}} ,\omega _k, 1\rangle $ . Let $D=\prod _{k\leq n} D_k$ , notice that as $\vec v$ represents a good $(n+2)$ -path we have $D=\mathfrak F(D)$ , hence $D \in \mathcal T_n$ . Furthermore, $\vec v_D =\vec v$ , hence the coding is onto as sought. As a Corollary we get that $|\mathcal T_n|=c_{n+2}$ .

In Figure 2 we present all good $4$ -paths and the corresponding types in $\mathcal T_2$ they encode.

Figure 2 All good $4$ -paths and the corresponding types in $\mathcal T_2$ they encode.

Lemma 3.1. Suppose $C,D\in \mathcal T_n$ and $\vec v_D \vartriangleleft \vec v_C$ , then $D\leq _T C$ .

Proof Let $D = \prod _{k\leq n} D_k$ and $C=\prod _{k\leq n} C_k$ . Note that if $D_k\leq _T C$ for every $k\leq n$ , then $D\leq _T C$ as sought. Fix $k\leq n$ , if $D_k=1$ , then clearly $D_k\leq C$ . Suppose $D_k\neq 1$ , we split to two cases:

$\blacktriangleright $ Suppose $C_k\neq 1$ . As $v^C_k <v^D_k$ and by (a) we have $D_k \leq _T C_k\leq _T C$ as sought.

$\blacktriangleright $ Suppose $C_k=1$ , let $m:=\max \{i\leq n \mid k< i,~v^C_i= v^C_k\}$ . As $v^C_i \leq i+1$ , by the coding m is well-defined and $v^C_m= v^C_k\leq k<m$ . Notice that $C_m=[\omega _m]^{<\omega _p}$ where $p=v^C_m$ and $D_k\equiv _T [\omega _k]^{<\omega _p}$ . So by (b), $D_k\leq _T C_m\leq _T C$ as sought.

Lemma 3.2. Suppose $C,D\in \mathcal T_n$ and $\vec v_D \not \vartriangleleft \vec v_C$ , then $D\not \leq _T C$ .

Proof Let $D=\prod _{k\leq n} D_k$ , $C=\prod _{k\leq n} C_k$ , $\vec v_C:=\langle v_0^C,\dots , v_n^C\rangle $ and $\vec v_D:=\langle v_0^D,\dots , v_n^D\rangle $ As $\vec v_D \not \vartriangleleft \vec v_C$ , we can define $i=\min \{k\leq n \mid v^C_k>v^D_k\}$ .

Let $p:=v^D_i$ and $r=\max \{k\leq n \mid v_i^D = v_k^D\}$ , notice that $p\leq i$ . We define a directed set F such that $F\leq _T D$ .

$\blacktriangleright $ Suppose $p=i$ and let $F=\omega _i$ . If $r=i$ , then clearly $F=D_i$ and $F\leq _T D$ as sought. Else, by the coding $D_r=[\omega _r]^{<\omega _p}$ . By Lemma 2.13, we have $F\leq _T D$ as sought.

$\blacktriangleright $ Suppose $p<i$ and let $F=\mathfrak D_{[\omega _i]^{<\omega _{p}}}$ . By the coding $D_r=[\omega _r]^{<\omega _p}$ and by Clause (b), we have $F\leq _T D$ as sought.

Notice that $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(F)) = \omega _p$ and $\operatorname {\mathrm {cf}}(F)=\omega _i$ . As $F\leq _T D$ , it is enough to verify that $F\not \leq _T C$ .

As $\vec v_C$ is a good $(n+2)$ -path, we know that $v_k^C>p$ for every $k\geq i$ . Consider $A:=\{i\leq k\leq n \mid C_k \neq 1 \}$ . We split to two cases:

$\blacktriangleright $ Suppose $A=\emptyset $ . Then $\operatorname {\mathrm {cf}}(\prod _{k\leq n} C_k) <\omega _i$ . As $\operatorname {\mathrm {cf}}(F)=\omega _i$ , by Lemma 2.12 we have that $F\not \leq _T \prod _{k\leq n} C_k$ as sought.

$\blacktriangleright $ Suppose $A\neq \emptyset $ . Let $E:=\prod _{k<i} C_k \times \prod _{k\in A} C_k$ Notice that $\operatorname {\mathrm {cf}}(\prod _{k<i} C_k)<\omega _i$ , $\prod _{i\leq k \leq n} C_k \equiv _T \prod _{k\in A} C_k$ and $C\equiv _T E$ . Furthermore, for each $k\in A$ , we have $\operatorname {\mathrm {non}}(\mathcal I_{bd}(C_k))>\omega _p$ . By Lemma 2.16, we have $\omega _p\in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E),\omega _i)$ . Recall $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(F)) = \omega _p$ . By Lemma 2.15, we get that $F\not \leq _T E$ , hence $F\not \leq _T C$ as sought.

Theorem 3.3. The posets $(\mathcal T_n, <_T)$ and $(\mathcal K_{n+2},\vartriangleleft )$ are isomorphic.

Proof Define f from $(\mathcal T_n,<_T)$ to $(\mathcal K_{n+2},\vartriangleleft )$ , where for $C\in \mathcal T_n$ , we let $f(C):=\vec v_C$ . By Lemmas 3.1 and 3.2, this is indeed an isomorphism of posets.

Furthermore, we claim that $\mathcal T_n$ contains one unique representative from each equivalence class of $(\mathcal S_n,\equiv _T)$ . Recall that the function $\mathfrak F$ is preserving Tukey equivalence classes. Consider two distinct $C,D\in \mathcal T_n$ . As the coding is a bijection, $\vec v_C$ and $\vec v_D$ are different. Notice that either $\vec v_C \not \vartriangleleft \vec v_D$ or $\vec v_D \not \vartriangleleft \vec v_C$ , hence by Lemma 3.2, $C\not \equiv _T D$ as sought.

Consider the poset $(\mathcal T_n, <_T)$ , clearly $1$ is a minimal element and by Lemma 2.13, $[\omega _n]^{<\omega }$ is a maximal element. By the previous theorem, the set of immediate successors of an element D in the poset $(\mathcal T_n, <_T)$ , is the set of all directed sets $C\in \mathcal T_n$ such that $\vec v_C$ is an $\vartriangleleft $ -immediate successor of $\vec v_D$ .

Lemma 3.4. Suppose $G,H\in \mathcal T_n$ , H is an immediate successor of G in the poset $(\mathcal T_n, <_T)$ and $(\vec v_G, \vec v_H)$ are on the l-diagonal. Then there are $C, E, M, N$ directed sets such that $:$

• • $G\equiv _T C\times M\times E$ and $H\equiv _T C\times N \times E$ ;

• • for some $k\leq n$ , $\operatorname {\mathrm {cf}}(N)=\omega _k$ , $|C|<\omega _k$ and either $E\equiv _T 1$ or $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))>\omega _{k-l}$ .

Furthermore,

• • If $l=0$ , then $M=1$ and $N=\omega _k$ .

• • If $l=1$ , then $k>1$ and $M=\omega _k$ and $N=[\omega _k]^{<\omega _{k-1}}$ .

• • If $l>1$ , then $k>l$ and $M=[\omega _k]^{<\omega _{k-l+1}}$ and $N=[\omega _k]^{<\omega _{k-l}}$ .

Proof As H is an immediate successor of G in the poset $(\mathcal T_n, <_T)$ , we know that $\vec v_H$ is an immediate successor of $\vec v_H$ in $(\mathcal K_{n+2},\vartriangleleft )$ . Let k be the unique $k\leq n$ such that $v^k_G = v^k_H +1$ .

Let $\vec v_G:=\langle v^0_G, \dots , v^n_G \rangle $ be a good $(n+2)$ -path coded by G. We construct $\langle M_i \mid i\leq n \rangle $ by letting $M_i$ be the ith element of the vector $\langle [\omega _i]^{<\omega },\dots , [\omega _i]^{<\omega _{i-1}} ,\omega _i, 1\rangle $ for every $i\leq n$ . Notice that $G\equiv _T \prod _{i\leq n} M_i$ . Similarly, we may construct $\langle N_i \mid i\leq n \rangle $ such that $H\equiv _T \prod _{i\leq n} N_i$ . Clearly, $M_i=N_i$ for every $i\neq k$ .

Let $C:=\prod _{i<k} M_i$ and $E:=\prod _{i>k} M_i$ . Notice that $|C|=\operatorname {\mathrm {cf}}(C)<\omega _k$ and either $E\equiv _T 1$ or $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))>\omega _{k-l}$ . Moreover, $G\equiv _T C\times M_k \times E$ and $H\equiv _TC\times N_k \times E$ . We split to cases:

• • If $l=0$ , then $v^k_H=k+1$ , hence $M_k=1$ and $N_k=\omega _k$ .

• • If $l=1$ , then $v^k_H=k$ , hence $M_k=\omega _k$ and $N_k=[\omega _k]^{<\omega _{k-1}}$ .

• • If $l>1$ , then $v^k_H=k-l+1$ , hence $M_k=[\omega _k]^{<\omega _{k-l+1}}$ and $N_k=[\omega _k]^{<\omega _{k-l}}$ .

Theorem 3.5. Suppose $G,H\in \mathcal T_n$ , H is an immediate successor of G in the poset $(\mathcal T_n, <_T)$ and $(\vec v_G, \vec v_H)$ are on the l-diagonal.

• • If $l=0$ , then there is no directed set $D\in \mathcal D_{\aleph _n}$ such that $G<_T D<_T H$ .

• • If $l>0$ , then consistently there exist a directed set $D\in \mathcal D_{\aleph _n}$ such that $G<_T D<_T H$ .

Proof Let $C,E, M, N$ be as in the previous lemma, so $G\equiv _T C\times M\times E$ , $H\equiv _T C\times N \times E$ and for some $k\leq n$ , $|C|<\omega _k$ and either $E\equiv _T 1$ or $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))>\omega _{k-l}$ . We split to three cases:

• • Suppose $l=0$ , then $G\equiv _T C\times E $ and $H\equiv _T C\times \omega _k \times E$ , by Theorem 4.1 there is no directed set D such that $G<_T D <_T H$ .

• • Suppose $l=1$ , then $k\geq 1$ and $N=[\omega _k]^{<\omega _{k-1}}$ and $M=\omega _k$ .

  1. – Suppose $k=1$ , then under the assumption $\mathfrak b =\omega _1$ , by Theorem 5.9 there exists a directed set D such that $G<_T D <_T H$ .

  2. – Suppose $k>1$ , then under the assumption $2^{\aleph _{k-2}} = \aleph _{k-1}$ and $2^{\aleph _{k-1}}= \aleph _k$ , by Corollary 5.1 there exists a directed set D such that $G<_T D <_T H$ .

• • Suppose $l>1$ , then $k\geq 2$ . Let $\theta =\omega _{k-l}$ and $\lambda = \omega _{k-1}$ . Notice $N=[\omega _k]^{\leq \theta }$ and $M=[\omega _k]^{<\theta }$ . In Corollary 5.11, we shall show that under the assumption $\lambda ^{\theta }<\lambda ^+$ and $ \clubsuit ^{\omega _{k-1}}_{J}(S,1) $ for some stationary set $S\subseteq E^{\omega _k}_{\theta }$ , there exists a directed set D such that $G<_T D <_T H$ .

4 Empty intervals in $D_{\aleph _n}$

Consider two successive directed sets in the poset $(\mathcal T_n , <_T)$ , we can ask whether there exists some other directed set in between in the Tukey order. The following theorem give us a scenario in which there is a no such directed set.

Theorem 4.1. Let $\kappa $ be a regular cardinal. Suppose C and E are two directed sets such that $\operatorname {\mathrm {cf}}(C)<\kappa $ and either $E\equiv _T 1$ or $\kappa \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E),\kappa )$ and $\kappa \leq \operatorname {\mathrm {cf}}(E)$ . Then there is no directed set D such that $C\times E <_T D <_T C\times \kappa \times E$ .

Proof By the upcoming Lemmas 4.2 and 4.3.

Lemma 4.2. Let $\kappa $ be a regular cardinal. Suppose C is a directed set such that $\operatorname {\mathrm {cf}}(C)<\kappa $ , then there is no directed set D such that $C <_T D <_T C \times \kappa $ .

Proof Suppose D is a directed set such that $C <_T D <_T C \times \kappa $ . Let us assume D is a directed set of size $\operatorname {\mathrm {cf}}(D)$ such that every subset of D of size $\operatorname {\mathrm {cf}}(D)$ is unbounded in D. By Lemma 2.12 we get that $\operatorname {\mathrm {cf}}(C) \leq \operatorname {\mathrm {cf}}(D)\leq \kappa $ . We split to two cases:

$\blacktriangleright $ Suppose $\operatorname {\mathrm {cf}}(C)\leq \operatorname {\mathrm {cf}}(D) <\kappa $ . Let $g:D\rightarrow C\times \kappa $ be a Tukey function. As $|D|=\operatorname {\mathrm {cf}}(D)<\kappa $ and $\kappa $ is regular there exists some $\alpha <\kappa $ such that $g"D \subseteq C\times \alpha $ . We claim that $\pi _C \circ g$ is a Tukey function from D to C, hence $D\leq _T C$ which is absurd. Suppose $X\subseteq D$ is unbounded in D, as g is a Tukey function, we know that $g"X$ is unbounded in $C\times \kappa $ . But as $(\pi _{\kappa }\circ g)"X$ is bounded by $\alpha $ , we get that $(\pi _{C}\circ g)" X$ is unbounded in C as sought.

$\blacktriangleright $ Suppose $\operatorname {\mathrm {cf}}(D)=\kappa $ , notice that $\kappa \in \operatorname {\mathrm {hu}}(D)$ is regular so by Lemma 2.9 we get that $\kappa \leq _T D$ . We also know that $C\leq _T D$ , thus $\kappa \times C \leq _T D$ which is absurd.

Note that $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))> \kappa $ implies that $\kappa \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E),\kappa )$ .

Lemma 4.3. Let $\kappa $ be a regular cardinal. Suppose C and E are two directed sets such that $\operatorname {\mathrm {cf}}(C)<\kappa \leq \operatorname {\mathrm {cf}}(E)$ and $\kappa \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E),\kappa )$ . Then there is no directed set D such that $C\times E <_T D <_T C\times \kappa \times E$ .

Proof Suppose D is a directed set such that $C\times E \leq _T D \leq _T C\times \kappa \times E$ , we will show that either $D\equiv _T C\times E$ or $D\equiv _T C\times \kappa \times E$ . We may assume that every subset of D of size $\operatorname {\mathrm {cf}}(D)$ is unbounded and $|C|=\operatorname {\mathrm {cf}}(C)$ . By Lemma 2.12, we have that $\operatorname {\mathrm {cf}}(E)=\operatorname {\mathrm {cf}}(D)$ .

Suppose first there exists some unbounded subset $X\in [D]^{\kappa }$ such that every subset $Y\in [X]^{<\kappa }$ is bounded. By Corollary 2.10, this implies that $\kappa \leq _T D$ . But as $C\times E\leq _T D$ and $D \leq _T C\times \kappa \times E$ , this implies that $C\times \kappa \times E \equiv _T D$ as sought.

Hereafter, suppose for every unbounded subset $X\in [D]^{\kappa }$ there exists some subset $Y\in [X]^{<\kappa }$ unbounded. Let $g:D\rightarrow C\times \kappa \times E$ be a Tukey function. Define $h:=\pi _{C\times E} \circ g$ . Now, there are two main cases to consider:

1. ▸ Suppose every unbounded subset $X\subseteq D$ of size $\lambda>\kappa $ which contain no unbounded subset of smaller cardinality is such that $h"X$ is unbounded in $C\times E$ .

   We show that h is Tukey, it is enough to verify that for every cardinal $\omega \leq \mu \leq \kappa $ and every unbounded subset $X\subseteq D$ of size $\mu $ which contain no unbounded subset of smaller cardinality is such that $h"X$ is unbounded in $C\times E$ .

   As g is Tukey, the set $g"X$ is unbounded in $C\times \kappa \times E$ . Notice that if the set $\pi _{C\times E}\circ g" X$ is unbounded, then we are done. Assume that $\pi _{C\times E}\circ g" X$ is bounded, then $\pi _{\kappa } \circ g"X$ is unbounded.

   $\blacktriangleright \blacktriangleright $ Suppose $|X|<\kappa $ . As $|g"X|<\kappa $ , we have that $\pi _{\kappa } \circ g"X$ is bounded, which is absurd.

   $\blacktriangleright \blacktriangleright $ Suppose $ |X|=\kappa $ , by the case assumption there exists some $Y \in [X]^{<\kappa }$ unbounded in D. But this is absurd as the assumption on X was that X contains no subset of size smaller than $|X|$ which is unbounded.

   $\blacktriangleright \blacktriangleright $ Suppose $|X|>\kappa $ , by the case assumption, $h"X$ is unbounded in $C\times E$ as sought.

2. ▸ Suppose for some unbounded subset $X\subseteq D$ of size $\lambda>\kappa $ which contains no unbounded subset of smaller cardinality is such that $h"X$ is bounded in $C\times E$ . As g is Tukey, $\pi _{\kappa }\circ g"X$ is unbounded.

   Let $X_\alpha := X\cap g^{-1} (C\times \{\alpha \}\times E )$ and $U_\alpha :=\bigcup _{\beta \leq \alpha } X_\beta $ for every $\alpha <\kappa $ . As g is Tukey and $g"U_\alpha $ is bounded, we get that $U_\alpha $ is also bounded by some $y_\alpha \in D$ . Let $Y:=\{y_\alpha \mid \alpha <\kappa \}$ . We claim that Y is of cardinality $\kappa $ . If it wasn’t, then by the pigeonhole principle as $\kappa $ is regular there would be some $\alpha <\kappa $ such that $y_\alpha $ bounds the set X in D and that is absurd. Similarly, as X is unbounded, the set Y and also every subset of it of size $\kappa $ must be unbounded.

   Next, we aim to get $Z\in [Y]^\kappa $ such that $\pi _{C\times E}\circ g"Z$ is bounded by some $(c,e)\in C\times E$ . This can be done as follows: As $|C|<\kappa $ and $\kappa $ is regular, by the pigeonhole principle, there exists some $Z_0\in [Y]^\kappa $ and $c\in C$ such that $g"Z_0 \subseteq \{c\}\times \kappa \times E$ . Similarly, if $|\pi _{E}\circ g" Z_0|<\kappa $ , by the pigeonhole principle, there exists some $Z\in [Z_0]^\kappa $ and $e\in E$ such that $g"Z \subseteq \{c\}\times \kappa \times \{e\}$ . Else, if $|\pi _{E}" Z_0|=\kappa $ , then as $\kappa \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E),\kappa )$ for some $B\in [\pi _{E}\circ g" Z_0]^{\kappa }$ and $e\in E$ , B is bounded in E by e. Fix some $Z\in [Z_0]^\kappa $ such that $g"Z\subseteq \{c\}\times \kappa \times B$ .

   Note that Z is a subset of Y of size $\kappa $ , hence, unbounded in D. By the assumption, there exists some subset $W\in [Z]^{<\kappa }$ unbounded in D. Note that as $\kappa $ is regular, for some $\alpha <\kappa $ , $\pi _{\kappa } \circ g"W\subseteq \alpha $ . As g is Tukey, the subset $g"W\subseteq \{c\}\times \kappa \times E$ is unbounded in $C\times \kappa \times E$ , but this is absurd as $g"W$ is bounded by $(c,\alpha ,e)$ .

5 Non-empty intervals

In this section we consider three types of intervals in the poset $(\mathcal T_n, <_T)$ and show each one can consistently have a directed set inside.

5.1 Directed set between $ \theta ^+ \times \theta ^{++}$ and $[\theta ^{++}]^{\leq \theta }$

In [6, Theorem 1.1], the authors constructed a directed set between $\omega _1\times \omega _2$ and $[\omega _2]^{\leq \omega }$ under the assumption $2^{\aleph _0}=\aleph _1$ , $2^{\aleph _1}=\aleph _2$ and the existence of an $\aleph _2$ -Souslin tree. In this subsection we generalize this result while waiving the assumption concerning the Souslin tree. The main corollary of this subsection is:

Corollary 5.1. Assume $\theta $ is an infinite cardinal such that $2^\theta = \theta ^+$ , $2^{\theta ^+}= \theta ^{++}$ . Suppose C and E are directed sets such that $\operatorname {\mathrm {cf}}(C)\leq \theta ^+$ and either $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}} (E))>\theta ^+$ or $E\equiv _T 1$ . Then there exists a directed set D such that $ C\times \theta ^+ \times \theta ^{++}\times E <_T C\times D\times E <_T C\times [\theta ^{++}]^{\leq \theta }\times E.$

The result follows immediately from Theorems 5.3 and 5.4. First, we prove the following required lemma.

Lemma 5.2. Suppose $\theta $ is a infinite cardinal and $D,J, E$ are three directed sets such that $:$

• • $\operatorname {\mathrm {cf}}(D)=\operatorname {\mathrm {cf}}(J)=\theta ^{++};$

• • $\theta ^+ \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D), \theta ^{++})$ and $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(J))\leq \theta ^+;$

• • $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))>\theta ^+$ or $E\equiv _T 1;$

• • $D\times E\leq _T J\times E$ .

Then $J\times E\not \leq _T D\times E$ . In particular, $D\times E <_T J\times E$ .

Proof Notice that D is a directed set such that every subset of size $\theta ^{++}$ contains a bounded subset of size $\theta ^+$ . Let us fix a cofinal subset $A\subseteq J$ of size $\theta ^{++}$ such that every subset of A of size $>\theta $ is unbounded in J.

Suppose on the contrary that $J\times E\leq _T D\times E$ . As $D\times E\leq _T J\times E$ we get that $D\times E\equiv _T J\times E$ , hence there exists some directed set X such that both $D\times E$ and $J\times E$ are cofinal subsets of X.

We may assume that D has an enumeration $D:=\{d_\alpha \mid \alpha <\theta ^{++}\}$ such that for every $\beta <\alpha <\theta ^{++}$ we have $d_\alpha \not <d_\beta $ . Fix some $e\in E$ . Now, for each $a\in A$ take a unique $x_a\in D$ and some $e_a\in E$ such that $(a,e)\leq _X (x_a,e_a)$ . To do that, enumerate $A=\{a_\alpha \mid \alpha <\theta ^{++}\}$ . Suppose we have constructed already the increasing sequence $\langle \nu _\beta \mid \beta <\alpha \rangle $ of elements in $\theta ^{++}$ . Pick some $\xi <\theta ^{++}$ above $\{ \nu _\beta \mid \beta <\alpha \}$ . As $D\times E$ is a directed set we may fix some $(x_{a_\alpha },e_a):=(d_{\nu _\alpha },e_a)\in D\times E$ above $(a_\alpha ,e)$ and $(d_\xi ,e)$ .

Set $T=\{x_a\mid a\in A\}$ , since $A\times E$ is cofinal in X, the set $T\times E$ is also cofinal in X and $D\times E$ . As $|T|=\theta ^{++}$ we get that there exists some subset $B\in [T]^{\theta ^+}$ bounded in D. Let $c\in D$ be such that $b\leq c$ for each $b\in B$ . Consider the set $K=\{a\in A \mid x_a \in B \}$ . Since either $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))>\theta ^+$ or $E\equiv _T 1$ , as $\{e_a\mid a\in K\}$ is of size $\leq \theta ^+$ , it is bounded in E by some $\tilde e\in E$ . So $P:=\{(x_a, e_a)\mid a\in K\}$ is bounded in X. Since B is of size $>\theta $ , the set K is also of size $>\theta $ . Thus, by the assumption on A, the set $K\times \{e\}$ is unbounded in $J\times E$ , but also in X because $J\times E$ is a cofinal subset of X. Then, for each $a\in K$ we have $(a,e)\leq _X (x_a,e_a) \leq _X (c,\tilde e)$ , contradicting the unboundedness of $K\times \{e\}$ in $J\times E$ .

Theorem 5.3. Suppose $\theta $ is an infinite cardinal and $C,D,E$ are directed sets such that $:$

1. (1) $\operatorname {\mathrm {cf}}(D)=\theta ^{++}$ .

2. (2) For every partition $D=\bigcup _{\gamma <\theta ^+} D_\gamma $ , there is an ordinal $\gamma <\theta ^+$ , and an unbounded $K\subseteq D_\gamma $ of size $\theta ^+$ .

3. (3) $\theta ^+ \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D), \theta ^{++})$ and $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))=\theta ^+$ .

4. (4) $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))>\theta ^+$ or $E\equiv _T 1$ .

5. (5) C is a directed set such that $\operatorname {\mathrm {cf}}(C)\leq \theta ^+$ .

Then $ C\times \theta ^+ \times \theta ^{++}\times E <_T C\times D\times E <_T C\times [\theta ^{++}]^{\leq \theta }\times E.$

Proof As $\operatorname {\mathrm {cf}}(D)=\theta ^{++}$ , we may assume that every subset of D of size $\theta ^{++}$ is unbounded.

Claim 5.3.1. $\theta ^+\times \theta ^{++}\leq _T D$ .

Proof As $\operatorname {\mathrm {cf}}(D)=\theta ^{++}$ , we get by Lemma 2.9 that $\theta ^{++}\leq _T D$ . Let K be an unbounded subset of D of size $\theta ^+$ , as every subset of size $\theta $ is bounded, by Corollary 2.10 we get that $\theta ^+\leq _T D$ . Hence, $\theta ^+\times \theta ^{++} \leq _T D$ as sought.

Claim 5.3.2. $D \leq _T [\theta ^{++}]^{\leq \theta }$ .

Proof As $\operatorname {\mathrm {cf}}(D)=\theta ^{++}$ and $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))=\theta ^+$ , by Lemma 2.13, $D \leq _T [\theta ^{++}]^{\leq \theta }$ as sought.

Notice this implies that $ C\times \theta ^+ \times \theta ^{++}\times E \leq _T C\times D\times E \leq _T C\times [\theta ^{++}]^{\leq \theta }\times E.$

By Lemma 2.17, as $|C\times \theta ^+|\leq \theta ^+$ , $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(\theta ^{++}\times E))>\theta ^+$ and Clause (2) we get that $D\not \leq _T C\times \theta ^+\times \theta ^{++}\times E$ .

Claim 5.3.3. $C\times [\theta ^{++}]^{\leq \theta }\times E \not \leq _T C\times D\times E$ .

Proof Recall that $\mathfrak D_{ [\theta ^{++}]^{\leq \theta }}\equiv _T [\theta ^{++}]^{\leq \theta }$ . Notice that following:

• • $\operatorname {\mathrm {cf}}(C\times D)=\operatorname {\mathrm {cf}}(C\times \mathfrak D_{[\theta ^{++}]^{\leq \theta }})=\theta ^{++}$ .

• • By Clause (3) we have that $\theta ^+ \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(C\times D), \theta ^{++})$ and $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}(C\times \mathfrak D_{ [\theta ^{++}]^{\leq \theta }}))\leq \theta ^{+}$ .

• • $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))>\theta ^+$ or $E=1$ .

• • $C\times D\times E\leq _T C\times \mathfrak D_{[\theta ^{++}]^{\leq \theta }}\times E$ .

So by Lemma 5.2 we are done.

We are left with proving the following theorem, in which we define a directed set $D_c$ using a coloring c.

Theorem 5.4. Suppose $\theta $ is an infinite cardinal such that $2^\theta = \theta ^+$ and $2^{\theta ^+}=\theta ^{++}$ . Then there exists a directed set D such that $:$

1. (1) $\operatorname {\mathrm {cf}}(D)=\theta ^{++}$ .

2. (2) For every partition $D=\bigcup _{\gamma <\theta ^+} D_\gamma $ , there is an ordinal $\gamma <\theta ^+$ , and an unbounded $K\subseteq D_\gamma $ of size $\theta ^+$ .

3. (3) $\theta ^+ \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D), \theta ^{++})$ and $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D))=\theta ^+$ .

The rest of this subsection is dedicated to proving Theorem 5.4. The arithmetic hypothesis will only play a role later on. Let $\theta $ be an infinite cardinal. For two sets of ordinals A and B, we denote $A\circledast B:= \{(\alpha ,\beta )\in A\times B\mid \alpha <\beta \}$ . Recall that by [3, Corollary 7.3], ${\mathsf {onto}}(\mathcal S,J^{\operatorname {\mathrm {bd}}}[\theta ^{++}],\theta ^+)$ holds for $\mathcal S:=[\theta ^{++}]^{\theta ^{++}}$ . This means that we may fix a coloring $c:[\theta ^{++}]^2\rightarrow \theta ^+$ such that for every $S\in \mathcal S$ and unbounded $B\subseteq \theta ^{++}$ , there exists $\delta \in S$ such that $c"(\{\delta \}\circledast B)=\theta ^+$ .

We fix some $S\in \mathcal S$ . For our purpose, it will suffice to assume that S is nothing but the whole of $\theta ^{++}$ . Let

$$ \begin{align*}D_{c}:=\{ X\in [\theta^{++}]^{\leq \theta^{+}} \mid \forall \delta\in S[ \{c(\delta,\beta) \mid \beta\in X\setminus(\delta+1)\}\in \operatorname{\mathrm{NS}}_{\theta^+} ]\}.\end{align*} $$

Consider $D_{c}$ ordered by inclusion, and notice that $D_{c}$ is a directed set since $\operatorname {\mathrm {NS}}_{\theta ^+}$ is an ideal.

Proposition 5.5. The following hold $:$

• • $[\theta ^{++}]^{\leq \theta } \subseteq D_{c} \subseteq [\theta ^{++}]^{\leq \theta ^+}$ .

• • $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D_c))\geq \operatorname {\mathrm {add}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D_c))\geq \theta ^+$ , i.e., every family of bounded subsets of $D_{c}$ of size $< \theta ^+$ is bounded.

• • If $2^{\theta ^+}=\theta ^{++}$ , then $|D_c|=\theta ^{++}$ , and hence $D_c\in \mathcal D_{\theta ^{++}}$ .

Lemma 5.6. For every partition $D_{c}=\bigcup _{\gamma <\theta ^+} D_\gamma $ , there is an ordinal $\gamma <\theta ^+$ , and an unbounded $E\subseteq D_\gamma $ of size $\theta ^+$ .

Proof As $[\theta ^{++}]^1$ is a subset of $D_{c}$ , the family $\{D_{\gamma }\mid \gamma <\theta ^+\}$ is a partition of the set $[\theta ^{++}]^1$ to at most $\theta ^+$ many sets. As $\theta ^+<\theta ^{++}=\operatorname {\mathrm {cf}}(\theta ^{++})$ , by the pigeonhole principle we get that for some $\gamma <\theta ^+$ and $b\in [\theta ^{++}]^{\theta ^{++}}$ , we have $[b]^1 \subseteq D_{\gamma }$ . Notice that by the assumption on the coloring c, there exists some $\delta \in S$ and $\delta <b'\in [b]^{\theta ^+}$ such that $c"(\delta \circledast b')=\theta ^+$ . Clearly the set $E:=[b']^1$ is a subset of $ D_{\gamma }$ of size $\theta ^+$ which is unbounded in $D_{c}$ .

Lemma 5.7. Suppose $2^{\theta }=\theta ^+$ , then $\theta ^+ \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D_{c}),\theta ^{++})$ .

Proof We follow the proof of [6, Lemma 5.4].

Let $D'$ be a subset of $D_{c}$ of size $\theta ^{++}$ we will show it contains a bounded subset of size $\theta ^+$ , let us enumerate it as $\{T_\gamma \mid \gamma <\theta ^{++}\}$ . Let, for each $X\in D_{c}$ and $\gamma \in S$ , $N^X_\gamma $ denote the non-stationary set $\{c(\gamma ,\beta )\mid \beta \in X\setminus (\gamma +1)\}$ , and let $G^X_\gamma $ denote a club in $\theta ^+$ disjoint from $N^X_\gamma $ .

As $2^{\theta }=\theta ^+$ we may fix a sufficiently large regular cardinal $\chi $ , and an elementary submodel $M\prec H_{\chi }$ of cardinality $\theta ^+$ containing all the relevant objects and such that $M^{\theta } \subseteq M$ . Denote $\delta =M\cap \theta ^{++}$ , notice $\delta \in E^{\theta ^{++}}_{\theta ^+}$ . Fix an increasing sequence $\langle \gamma _\xi \mid \xi <\theta ^+\rangle $ in $\delta $ such that $\sup \{\gamma _\xi \mid \xi <\theta ^+\}=\delta $ . Enumerate $\delta \cap S=\{s_\xi \mid \xi <\theta ^+\}$ . In order to simplify notation, let $G^\gamma _\xi $ denote the set $G^{T_\gamma }_{s_\xi }$ for each $\gamma <\theta ^{++}$ and $\xi <\theta ^+$ .

We construct by recursion on $\xi <\theta ^+$ three sequences $\langle \delta _\xi \mid \xi <\theta ^+\rangle $ , $\langle \Gamma _\xi \mid \xi <\theta ^+\rangle $ and $\langle \eta _\xi \mid \xi <\theta ^+\rangle $ with the following properties:

1. (1) $\langle \delta _\xi \mid \xi <\theta ^+\rangle $ is an increasing sequence converging to $\delta $ .

2. (2) $\langle \Gamma _\xi \mid \xi <\theta ^+ \rangle $ is a decreasing $\subseteq $ -chain of stationary subsets of $\theta ^{++}$ each one containing $\delta $ and definable in M.

3. (3) $\langle \eta _\xi \mid \xi <\theta ^+ \rangle $ is an increasing sequence of ordinals below $\theta ^+$ .

4. (4) $G^\delta _{\zeta } \cap \eta _{\mu } = G^{\delta _{\mu }}_{\zeta }\cap \eta _{\mu } $ for $\zeta \leq \mu <\theta ^+$ .

$\blacktriangleright $ Base case: Let $\eta _0$ be the first limit point of $G^\delta _0$ . Notice that $G^\delta _0\cap \eta _0$ is an infinite set of size $\leq \theta $ below $\delta $ , hence it is inside of M. Let

$$ \begin{align*}\Gamma_0:=\{\gamma<\theta^{++} \mid G^\delta_0\cap \eta_0 = G^\gamma_0\cap \eta_0\}.\end{align*} $$

Since $\delta \in \Gamma _0$ , the set $\Gamma _0$ is stationary in $\theta ^{++}$ . Let $\delta _0:=\min (\Gamma _0)$ .

$\blacktriangleright $ Suppose $\xi _0<\theta ^+$ , and that $\delta _\xi $ , $\Gamma _\xi $ and $\eta _\xi $ have been constructed for each $\xi <\xi _0$ . Let $\eta _{\xi _0}$ be the first limit point of $G^\delta _{\xi _0}\setminus \sup \{\eta _\xi \mid \xi <\xi _0\}$ . Consider the set

$$ \begin{align*}\Gamma_{\xi_0}=\large \{\gamma \in \bigcap_{\xi<\xi_0}\Gamma_\xi \mid \forall \xi\leq \xi_0 [G^\delta_\xi\cap \eta_{\xi_0} = G^\gamma_{\xi}\cap \eta_{\xi_0}]\large \}.\end{align*} $$

Since $\Gamma _{\xi _0}$ belongs to M, and since $\delta \in \Gamma _{\xi _0}$ , it must be that $\Gamma _{\xi _0}$ is stationary in $\theta ^{++}$ . Since $\Gamma _{\xi _0}$ is cofinal in $\theta ^{++}$ and belongs to M, the set $\delta \cap \Gamma _{\xi _0}$ is cofinal in $\delta $ . Define $\delta _{\xi _0}$ be the minimal ordinal in $\delta \cap \Gamma _{\xi _0}$ greater than both $\sup \{\delta _\xi \mid \xi <\xi _0\}$ and $\gamma _{\xi _0}$ . It is clear from the construction that conditions $(1-4)$ are satisfied.

The following claim gives us the wanted result.

Claim 5.7.1. The set $\{T_{\delta _\xi }\mid \xi <\theta ^+\}$ is a subset of $D'$ of size $\theta ^+$ which is bounded in $D_{c}$ .

Proof As the order on $D_{c}$ is $\subseteq $ , it suffices to prove that the union $T=\bigcup _{\xi <\theta ^+} T_{\delta _\xi }\in D_{c}$ . Since, for each $\xi <\theta ^+$ , both $\delta _\xi $ and $\langle T_\gamma \mid \gamma <\theta ^{++} \rangle $ belong to M, it must be that $T_{\delta _\xi }\in M$ . Since $\theta ^+\in M$ and $M\models |T_{\delta _\xi }|\leq \theta ^+$ , we have $T_{\delta _\xi }\subseteq M$ . Thus $T\subseteq M$ and furthermore $T\subseteq \delta $ . This means that, in order to prove that $T\in D_{c}$ , it is enough to prove that for each $t \in S\cap \delta $ , the set $\{c(t,\beta )\mid \beta \in T\setminus (t+1)\}$ is non-stationary in $\theta ^+$ . Fix some $t\in S\cap \delta $ . Let $\zeta <\theta ^+$ be such that $s_\zeta = t$ . Define

$$ \begin{align*}G:=G^\delta_\zeta \cap (\bigcap_{\xi\leq \zeta} G^{\delta_\xi}_\zeta)\cap (\triangle_{\xi<\theta^+} G^{\delta_\xi}_\zeta).\end{align*} $$

Since the intersection of $< \theta ^+$ -many clubs in $\theta ^+$ is a club, and since diagonal intersection of $\theta ^+$ many clubs is a club, we know that G is a club in $\theta ^+$ .

We will prove that $G\cap \{c(t,\beta )\mid \beta \in T\setminus (t+1)\}=\emptyset $ . Suppose $\alpha <\theta ^+$ is such that $\alpha \in G \cap \{c(t,\beta )\mid \beta \in T\setminus (t+1)\}$ . This means that $\alpha \in G$ and that for some $\mu <\theta ^+$ and $\beta \in T_{\delta _\mu }\setminus (t+1)$ we have $\alpha = c(t,\beta )$ . So $\alpha \in N^{T_{\delta _\mu }}_{t}$ . Note that this implies that $\alpha \notin G^{\delta _\mu }_\zeta $ . Let us split to three cases:

$\blacktriangleright $ Suppose $\mu \leq \zeta $ , then since $\alpha \in \bigcap _{\xi \leq \zeta } G^{\delta _\xi }_\zeta $ , we have that $\alpha \in G^{\delta _\mu }_\zeta $ which is clearly contradicting $\alpha \notin G^{\delta _\mu }_\zeta $ .

$\blacktriangleright $ Suppose $\mu>\zeta $ and $\alpha <\eta _\mu $ . Then by $(4)$ , we have that $G^\delta _\zeta \cap \eta _\mu =G^{\delta _\mu }_\zeta \cap \eta _\mu $ . As $\alpha \not \in G^{\delta _\mu }_\zeta $ and $\alpha <\eta _\mu $ , it must be that $\alpha \notin G^\delta _\zeta $ . Recall that $\alpha \in G$ , but this is absurd as $G\subseteq G^\delta _\zeta $ and $\alpha \notin G^\delta _\zeta $ .

$\blacktriangleright $ Suppose $\mu>\zeta $ and $\alpha \geq \eta _\mu \geq \mu $ . As $\alpha \in G$ , we have that $\alpha \in \triangle _{\xi <\theta ^+} G^{\delta _\xi }_\zeta $ . As $\alpha>\mu $ , we get that $\alpha \in G^{\delta _\mu }_\zeta $ which is clearly contradicting $\alpha \notin G^{\delta _\mu }_\zeta $ .

5.2 Directed set between $\omega \times \omega _1$ and $[\omega _1]^{<\omega }$

As mentioned in [8], by the results of Todorčević [13], it follows that under the assumption $\mathfrak b=\omega _1$ there exists a directed set of size $\omega _1$ between the directed sets $\omega \times \omega _1$ and $[\omega _1]^{<\omega }$ . In this subsection we spell out the details of this construction.

For two functions $f,g\in {}^{\omega }\omega $ , we define the order $<^*$ by $f<^*g$ iff the set $\{n<\omega \mid g(n)\geq f(n)\}$ is finite. Furthermore, by $f\lhd g$ we means that there exists $m<\omega $ such that for all $n<m$ we have $f(n)\leq g(n)$ and $f(k)<g(k)$ whenever $m\leq k<\omega $ . Assuming $f\leq ^* g$ , we let $\Delta (f,g):=\min \{m<\omega \mid \forall n\geq m [f(n)\leq g(n)]\}$ .

The following fact is a special case of [13, Theorem 1.1] in the case $n=0$ , for complete details we give the proof as suggested by the referee.

Fact 5.8 (Todorčević [13, Theorem 1.1]).

Suppose A is an uncountable sequence of ${}^{\omega }\omega $ of increasing functions which are $<^*$ -increasing and $\leq ^*$ -unbounded, then there are $f,g\in A$ such that $f\lhd g$ .

Proof Let $A:=\{g_\alpha \mid \alpha <\omega _1\}$ be an uncountable sequence of increasing functions of ${}^{\omega }\omega $ which are $<^*$ -increasing and $\leq ^*$ -unbounded.

Let us fix a countable elementary sub-model $M\prec (H_{\omega _2},\in )$ with $A\in M$ . Let $\delta :=\omega _1\cap M$ , $B:=\omega _1\setminus (\delta +1)$ and write $B_n:=\{\beta \in B\mid \Delta (g_\delta ,g_\beta )=n \}$ . As $B=\bigcup _{n<\omega }B_n$ , let us fix some $n<\omega $ such that $B_n$ is uncountable. As $\{g_\alpha \mid \alpha \in B_n\}$ is unbounded, we get that the set $K:=\{m<\omega \mid \sup \{g_\beta (m)\mid \beta \in B_n\} = \omega \}$ is non-empty, so consider the minimal element, $m:=\min (K)$ . For $t\in {}^{m}\omega $ , denote $B^t_n:=\{\beta \in B_n \mid t\subseteq g_\beta \}$ . By minimality of m, the set $\{ t\in {}^{m}\omega \mid B^t_n \neq \emptyset \}$ is finite, so we can easily find some $t\in {}^{m}\omega $ such that $\sup \{ g_\beta (m)\mid \beta \in B^t_n\}=\omega $ .

Note that the set $\{\beta < \omega _1 \mid t\subseteq g_\beta \}$ is a non-empty set that is definable from A and t, hence it is in M. Let us fix some $\alpha \in M\cap \omega _1$ such that $t\subseteq g_\alpha $ . Put $k:=\Delta (g_\alpha ,g_\delta )$ , and then pick $\beta \in B^t_n$ such that $g_\beta (m)>g_\alpha (k+n)$ . Of course, $\alpha <\delta <\beta $ . We claim that $g_\alpha \vartriangleleft g_\beta $ as sought.

Let us divide to three cases:

• • If $i<m$ , then $g_\alpha (i)=t(i)=g_\beta (i)$ .

• • If $m\leq i\leq k+n$ , then $g_\alpha (i)\leq g_\alpha (k+n)<g_\beta (m)\leq g_\beta (i)$ recall that every function in A is increasing.

• • If $k+n<i<\omega $ , then $\Delta (g_\alpha ,g_\delta )=k<i$ and $g_\alpha (i)\leq g_\delta (i)$ , as well as $\Delta (g_\delta ,g_\beta )=n<i$ and $g_\delta (i)\leq g_\beta (i)$ . Altogether, $g_\alpha (i)\leq g_\beta (i)$ .

Theorem 5.9. Assume $\mathfrak b =\omega _1$ . Suppose E is a directed set such that $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}} (E))>\omega $ or $E\equiv _T 1$ . Then there exists a directed set D such that

$$ \begin{align*}\omega\times \omega_1 \times E <_T D\times E <_T [\omega_1]^{<\omega}\times E.\end{align*} $$

Proof Let $\mathcal F:=\langle f_\alpha \mid \alpha <\omega _1 \rangle \subseteq {}^{\omega }\omega $ witness $\mathfrak b=\omega _1$ . Recall $\mathcal F$ is a $<^*$ -increasing and unbounded sequence, i.e., for every $g\in {}^{\omega }\omega $ , there exists some $\alpha <\omega _1$ such that $f_\beta \not \leq ^* g$ , whenever $\alpha <\beta <\omega _1$ .

For a finite set of functions $F\subseteq {}^{\omega }\omega $ , we define a function $h:=\max (F)$ which is $\lhd $ -above every function in F by letting $h(n):=\max \{f(n)\mid f\in F\}$ . We consider the directed set $D:=\{ \max (F)\mid F\subseteq \mathcal F,~ |F|<\aleph _0 \}$ , ordered by the relation $\lhd $ , clearly D is a directed set.

Claim 5.9.1. Every uncountable subset $X\subseteq D$ contains a countable $B\subset X$ which is unbounded in D.

Proof Let X be an uncountable subset of D. As $\mathcal F$ is a $<^*$ -increasing and unbounded, also X contains an uncountable $<^*$ -unbounded subset $Y\subseteq X$ . As no function $g:\omega \rightarrow \omega $ is $<^*$ -bounding the set Y, we can find an infinite countable subset $B\subseteq Y$ and $n<\omega $ such that $\{f(n)\mid f\in B \}$ is infinite. Clearly B is $\lhd $ -unbounded in D as sought.

Claim 5.9.2. $\omega \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D),\omega _1) $ .

Proof We show that every uncountable subset of D contains a countable infinite bounded subset. Let $A\subseteq D$ be an uncountable set, we may refine A and assume that it is $<^*$ -increasing and unbounded. We enumerate $A:=\{g_\alpha \mid \alpha <\omega _1\}$ and define a coloring $c:[\omega _1]^2\rightarrow 2$ , letting for $\alpha <\beta <\omega _1$ the color $c(\alpha ,\beta )=1$ iff $g_\alpha \lhd g_\beta $ . Recall that Erdös and Rado showed that $\omega _1\rightarrow (\omega _1,\omega +1)^2$ , so either there is an uncountable homogeneous set of color $0$ or there exists an homogeneous set of color $1$ of order-type $\omega +1$ . Notice that Fact 5.8 contradicts the first alternative, so the second one must hold. Let $X\subseteq \omega _1$ be a set such that $\operatorname {\mathrm {otp}}(X)=\omega +1$ and $c"[X]^2=\{1\}$ , notice that $\{g_\alpha \mid \alpha \in X\}$ is an infinite countable subset of A which is $\lhd $ -bounded by the function $g_{\max (X)}\in A$ as sought.here

Note that $\operatorname {\mathrm {cf}}(D)=\omega _1$ , hence $D\times E \leq _T [\omega _1]^{<\omega }\times E$ .

Claim 5.9.3. $\omega \times \omega _1\times E\leq _T D\times E$ .

Proof As every subset of D of size $\omega _1$ is unbounded, we get by Lemma 2.9 that $\omega _1\leq _T D$ . As D is a directed set, every finite subset of D is bounded. By Claim 5.9.1, D contains an infinite countable unbounded subset, so by Corollary 2.10 we have $\omega \leq _T D$ . Finally, $\omega \times \omega _1 \leq _T D$ as sought.

Claim 5.9.4. $D\not \leq _T\omega \times E$ .

Proof Recall that either $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}} (E))>\omega $ or $E\equiv _T 1$ . Note that if $E\equiv _T 1$ , then as $\operatorname {\mathrm {cf}}(D)=\omega _1>\operatorname {\mathrm {cf}}(\omega )$ , we have by Lemma 2.12 that $D\not \leq _T\omega \times E$ as sought. Note that for every partition $D=\bigcup \{D_n\mid n<\omega \}$ of D, there exists some $n<\omega $ such that $D_n$ is uncountable, and by Claim 5.9.1, there exists some $X\subseteq D_n$ infinite and unbounded in D. As $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}} (E))>\omega $ , by Lemma 2.17 we have $D\not \leq _T\omega \times E$ as sought.

Claim 5.9.5. $[\omega _1]^{<\omega } \not \leq _T D\times E$ .

Proof By Claim 5.9.2, every uncountable subset of D contains an infinite countable bounded subset and every countable subset of $E $ is bounded, we get that $\omega \in \operatorname {\mathrm {in}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D\times E),\omega _1)$ . As $\operatorname {\mathrm {out}}(\mathcal I_{\operatorname {\mathrm {bd}}}([\omega _1]^{<\omega }))=\omega $ by Lemma 2.15 we get that $[\omega _1]^{<\omega } \not \leq _T D\times E$ as sought.

5.3 Directed set between $[\lambda ]^{<\theta }\times [\lambda ^+]^{\leq \theta }$ and $[\lambda ^+]^{< \theta }$

In [6, Theorem 1.2], the authors constructed a directed set between $[\omega _1]^{<\omega }\times [\omega _2]^{\leq \omega }$ and $[\omega _2]^{< \omega }$ under the assumption $2^{\aleph _0}=\aleph _1$ , $2^{\aleph _1}=\aleph _2$ and the existence of a non-reflecting stationary subset of $E^{\omega _2}_\omega $ . In this subsection we generalize this result while waiving the assumption concerning the non-reflecting stationary set.

We commence by recalling some classic guessing principles and introducing a weak one, named $ \clubsuit ^\mu _{J}(S,1) $ , which will be useful for our construction.

Definition 5.10. For a stationary subset $ S\subseteq \kappa $ :

1. (1) $ \diamondsuit (S) $ asserts the existence of a sequence $ \langle C_\alpha \mid \alpha \in S \rangle $ such that:

   • • for all $ \alpha \in S $ , $ C_\alpha \subseteq \alpha $ ;

   • • for every $B\subseteq \kappa $ , the set $\{\alpha \in S\mid B\cap \alpha =C_\alpha \}$ is stationary.

2. (2) $ \clubsuit (S) $ asserts the existence of a sequence $ \langle C_\alpha \mid \alpha \in S \rangle $ such that:

   • • for all $ \alpha \in S\cap \operatorname {\mathrm {acc}}(\kappa ) $ , $ C_\alpha $ is a cofinal subset of $\alpha $ of order type $\operatorname {\mathrm {cf}}(\alpha )$ ;

   • • for every cofinal subset $ B\subseteq \kappa $ , the set $\{\alpha \in S \mid C_\alpha \subseteq B \}$ is stationary.

3. (3) $ \clubsuit ^\mu _{J}(S,1) $ asserts the existence of a sequence $ \langle C_\alpha \mid \alpha \in S \rangle $ such that:

   • • for all $ \alpha \in S\cap \operatorname {\mathrm {acc}}(\kappa ) $ , $ C_\alpha $ is a cofinal subset of $\alpha $ of order type $\operatorname {\mathrm {cf}}(\alpha )$ ;

   • • for every partition $\langle A_\beta \mid \beta <\mu \rangle $ of $\kappa $ there exists some $\beta <\mu $ such that the set $\{\alpha \in S \mid \sup (C_\alpha \cap A_\beta )=\alpha \}$ is stationary.

Recall that by a Theorem of Shelah [10], for every uncountable cardinal $\lambda $ which satisfy $2^\lambda = \lambda ^+$ and every stationary $S\subseteq E^{\lambda ^+}_{\neq \operatorname {\mathrm {cf}}(\lambda )}$ , $\Diamond (S)$ holds. It is clear that $\Diamond (S) \Rightarrow \clubsuit (S) \Rightarrow \clubsuit ^\lambda _{J}(S,1) $ . The main corollary of this subsection is:

Corollary 5.11. Let $\theta <\lambda $ be two regular cardinals. Assume $\lambda ^{\theta }<\lambda ^+$ and $\clubsuit ^\lambda _{J}(S,1) $ holds for some stationary $S\subseteq E^{\lambda ^+}_\theta $ . Suppose C and E are two directed sets such that $\operatorname {\mathrm {cf}}(C)<\lambda ^+$ and $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}} (E))>\theta $ or $E\equiv _T 1$ . Then there exists a directed set $D_{\mathcal C}$ such that $:$

$$ \begin{align*}C\times [\lambda]^{<\theta}\times [\lambda^+]^{\leq \theta}\times E<_T C\times [\lambda]^{<\theta}\times D_{\mathcal C} \times E <_T C\times [\lambda^+]^{< \theta} \times E.\end{align*} $$

In the rest of this subsection we prove this result.

Suppose $\mathcal C:=\langle C_\alpha \mid \alpha \in S \rangle $ is a C-sequence for some stationary set $S\subseteq E^{\lambda ^+}_\theta $ , i.e., $C_\alpha $ is a cofinal subset of $\alpha $ of order-type $\theta $ , whenever $\alpha \in S$ . We define the directed set $D_{\mathcal C} :=\{ Y\in [\lambda ^+]^{\leq \theta }\mid \forall \alpha \in S [|Y\cap C_\alpha |<\theta ] \}$ ordered by $\subseteq $ . Notice that $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(D_{\mathcal C}))= \theta $ and $[\lambda ^+]^{< \theta }\subseteq D_{\mathcal C}$ .

Recall that by Hausdorff’s formula $(\lambda ^+)^\theta = \max \{\lambda ^+,\lambda ^\theta \}$ , so if $\lambda ^{\theta }<\lambda ^+$ , then $(\lambda ^+)^\theta =\lambda ^+$ . So we may assume $|D_{\mathcal C}|=\lambda ^+$ .

Claim 5.11.1. Suppose $|D_{\mathcal C}|=\lambda ^+$ , then $[\lambda ^+]^{\leq \theta } \leq _T D_{\mathcal C}$ .

Proof Fix a bijection $\phi :D_{\mathcal C}\rightarrow \lambda ^+$ . Denote $X:=\{x\cup \{\phi (x)\}\mid x\in D_{\mathcal C}\}$ , clearly X is cofinal subset of $D_{\mathcal C}$ . Let us fix some injective function $g:[\lambda ^+]^{\leq \theta } \rightarrow X$ . We claim that g is a Tukey function, which witness that $[\lambda ^+]^{\leq \theta } \leq _T D_{\mathcal C}$ . Fix some $B\subseteq [\lambda ^+]^{\leq \theta }$ unbounded in $ [\lambda ^+]^{\leq \theta }$ , note that $|B|>\theta $ . As g is injective, we get that $g"B$ is a set of size $>\theta $ . Notice that there exists $Z\in [\lambda ^+]^{\theta ^+}$ such that $Z\subseteq \bigcup g"B$ . Assume that $g"B$ is bounded by $d\in D_{\mathcal C}$ in $D_{\mathcal C}$ . As $D_{\mathcal C}$ is ordered by $\subseteq $ , we get that $Z\subseteq d$ , so $|d|\geq \theta ^+$ . But this is a absurd as every set in $D_{\mathcal C}$ is of size $\leq \theta $ .

Notice that by Lemma 2.13 and Claim 5.11.1, as $(\lambda ^+)^\theta =\lambda ^+$ we have $ [\lambda ]^{<\theta }\times [\lambda ^+]^{\leq \theta } \leq _T [\lambda ]^{<\theta } \times D_{\mathcal C} \leq _T [\lambda ^+]^{< \theta }.$ Hence, $C\times [\lambda ]^{<\theta }\times [\lambda ^+]^{\leq \theta }\times E\leq _T C\times [\lambda ]^{<\theta } \times D_{\mathcal C}\times E \leq _T C\times [\lambda ^+]^{< \theta } \times E.$

Claim 5.11.2. Suppose $\mathcal C$ is a $ \clubsuit ^\lambda _{J}(S,1) $ -sequence and $:$

1. (i) C is a directed set such that $|C|<\lambda ^+$ ;

2. (ii) E is a directed set such that $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}}(E))>\theta $ and $\operatorname {\mathrm {cf}}(E)\geq \lambda ^+$ .

Then $ C\times D_{\mathcal C} \not \leq _T C\times E$ .

Proof Suppose that $f: C\times D_{\mathcal C} \rightarrow C\times E$ is a Tukey function. Fix some $o\in C$ and for each $\xi <\lambda ^+$ , denote $(c_\xi , x_\xi ) := f(o,\{\xi \})$ . Consider the set $\{(c_\xi ,x_\xi ) \mid \xi <\lambda ^+\}$ . For every $c\in C$ , we define $A_c:=\{\xi <\lambda ^+ \mid c_\xi =c\}$ , clearly $\langle A_c \mid c\in C \rangle $ is a partition of $\lambda ^+$ to less than $\lambda ^+$ many sets.

As $\mathcal C$ is a $ \clubsuit ^\lambda _{J}(S,1) $ -sequence, there exists some $c\in C$ and $\alpha \in S$ such that $|C_\alpha \cap A_c|=\theta $ . Let us fix some $B\in [C_\alpha \cap A_c]^{\theta }$ . Notice that the set $G:=\{(o,\{\xi \})\mid \xi \in B\}$ is unbounded in $ C\times D_{\mathcal C}$ , hence as f is Tukey, $f"G$ is unbounded in $C\times E$ . The subset $\{x_\xi \mid \xi \in B\}$ of E is of size $\theta $ , hence bounded by some e. Note that $f"G=\{( c,x_\xi )\mid \xi \in B\}$ is bounded by $(c,e)$ in $ C\times E$ which is absurd.

By the previous claim, as $\lambda ^\theta <\lambda ^+$ , we get that $ C\times D_{\mathcal C}\times [\lambda ]^{<\theta } \times E \not \leq _T C\times [\lambda ]^{<\theta }\times [\lambda ^+]^{\leq \theta }\times E $ . The following claim gives a negative answer to the question of whether there is a C-sequence $\mathcal C$ such that $D_{\mathcal C} \equiv _T [\lambda ^+]^{<\theta }$ .

In the following claim we use the fact that the sets in the sequence $\mathcal C$ are of a bounded cofinality.

Claim 5.11.3. Assume $\lambda ^\theta <\lambda ^+$ . Suppose $S\subseteq E^{\lambda ^+}_\theta $ is a stationary set and $\mathcal C:=\langle C_\alpha \mid \alpha \in S \rangle $ is a C-sequence, then $D_{\mathcal C}\not \geq _T [\lambda ^+]^{<\theta }$ .

Proof Let $S\subseteq E^{\lambda ^+}_\theta $ and $\mathcal C:=\langle C_\alpha \mid \alpha \in S \rangle $ be a C-sequence. Suppose we have $ [\lambda ^+]^{< \theta } \leq _T D_{\mathcal C} $ , let $f:[\lambda ^+]^{< \theta } \rightarrow D_{\mathcal C}$ be a Tukey function and $Y:=f"[\lambda ^+]^{1}$ . Let us split to two cases:

$\blacktriangleright $ Suppose $|Y|<\lambda ^+$ . By the pigeonhole principle, we can find a subset $Q\subseteq [\lambda ]^1$ of size $\theta $ such that $f"Q=\{x\}$ for some $x\in D_{\mathcal C}$ . As f is Tukey and Q is unbounded in $[\lambda ^+]^{< \theta }$ , the set $f"Q$ is unbounded which is absurd.

$\blacktriangleright $ Suppose $|Y|=\lambda ^+$ . As f is Tukey, every subset of Y of size $\theta $ is unbounded which is absurd to the following claim.

Subclaim 5.11.3.1. There is no subset $Y\subseteq D_{\mathcal C}$ of size $\lambda ^+$ such that every subset of Y of size $\theta $ is unbounded.

Proof Assume towards a contradiction that Y is such a set. As $\lambda ^{\theta }<\lambda ^+$ , we may refine Y and assume that $Y=\{y_\alpha \mid \alpha <\lambda ^+\}$ is a $\Delta $ -system with a root R separated by a club $C\subseteq \lambda ^+$ , i.e., such that for every $\alpha <\beta <\lambda ^+$ , $y_\alpha \setminus R <\eta < y_\beta \setminus R$ for some $\eta \in C$ .

We define an increasing sequence of ordinals $\langle \beta _\nu \mid \nu \leq \theta ^2\rangle $ where for each $\nu \leq \theta ^2$ we let $\beta _\nu :=\sup \{y_\xi \mid \xi <\nu \}$ . As C is a club, we get that $\beta _{\theta \cdot \nu }\in C$ for each $\nu <\theta $ .

We aim to construct a subset $X=\{x_j \mid j <\theta \}$ of Y, we split to two cases: Suppose $\beta _{\theta ^2}\in S$ . Recall that $\operatorname {\mathrm {otp}} (C_{\beta _{\theta ^2}})=\theta $ and $\sup (C_{\beta _{\theta ^2}})=\beta _{\theta ^2}$ , so for every $j <\theta $ we have that the interval $[\beta _{\theta \cdot j},\beta _{\theta \cdot (j+1)})$ contains $<\theta $ many elements of the ladder $C_{\beta _{\theta ^2}}$ , let us fix some $x_j\in Y$ such that $x_j\setminus R\subset [\beta _{\theta \cdot j},\beta _{\theta \cdot (j+1)})$ and $x_j\setminus R$ is disjoint from $C_{\beta _{\theta ^2}}$ . If $\beta _{\theta ^2}\notin S$ , define $X:=\{x_j \mid j<\theta \}$ where $x_j:= y_{\theta \cdot j}$ .

Let us show that $X=\{x_j \mid j <\theta \}$ is a bounded subset of Y, which is a contradiction to the assumption. It is enough to show that for every $\alpha \in S$ , we have that $|(\bigcup X)\cap C_\alpha |<\theta $ . Let $\alpha \in S$ .

$\blacktriangleright $ Suppose $\alpha>\beta _{\theta ^2}$ , as $C_\alpha $ is a cofinal subset of $\alpha $ of order-type $\theta $ and $\bigcup X$ is bounded by $\beta _{\theta ^2}$ it is clear that $|(\bigcup X)\cap C_\alpha |<\theta $ .

$\blacktriangleright $ Suppose $\alpha <\beta _{\theta ^2}$ . As $C_\alpha $ if cofinal in $\alpha $ and of order-type $\theta $ , there exists some $j <\theta $ such that for all $j <\rho <\theta $ , we have $(x_\rho \setminus R)\cap C_\alpha =\emptyset $ . As $x_\rho \in D_{\mathcal C_R}$ for every $\rho <\theta $ and $\theta $ is regular, we get that $|(\bigcup X)\cap C_\alpha |<\theta $ as sought.

$\blacktriangleright $ Suppose $\alpha = \beta _{\theta ^2}$ . Notice this implies that we are in the first case of the construction of the set X. Recall that the $\Delta $ -system $\{x_j \mid j <\theta \}$ is such that $(x_j\setminus R )\cap C_\alpha =\emptyset $ , hence $(\bigcup X)\cap C_{\alpha } = R \cap C_\alpha $ . Recall that as $x_0\in D_{\mathcal C}$ , we get that $R \cap C_\alpha $ is of size $<\theta $ , hence also $(\bigcup X)\cap C_{\alpha }$ is as sought.

Claim 5.11.4. Assume $\lambda ^\theta <\lambda ^+$ . Suppose C and E are two directed sets such that $|C|<\lambda ^+$ and either $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}} (E))>\theta $ or $E\equiv _T 1$ . Then for every C-sequence $\mathcal C$ on a stationary $S\subseteq E^{\lambda ^+}_\theta $ , $C\times [\lambda ^+]^{<\theta } \times E\not \leq _T C\times D_{\mathcal C} \times E$ .

Proof Let $\mathcal C:=\langle C_\alpha \mid \alpha \in S \rangle $ be a C-sequence where $S\subseteq E^{\lambda ^+}_\theta $ . Suppose on the contrary that $C\times [\lambda ^+]^{<\theta } \times E \leq _T C\times D_{\mathcal C} \times E$ . Hence, $[\lambda ^+]^{<\theta }\leq _T C\times D_{\mathcal C} \times E$ , let us fix a Tukey function $f:[\lambda ^+]^{<\theta }\rightarrow C\times D_{\mathcal C}\times E$ witnessing that. Consider $X=[\lambda ^+]^1$ .

By the pigeonhole principle, there exists some $c\in C$ and some set $Z\subseteq X$ of size $\lambda ^+$ such that $f"Z \subseteq \{ c\}\times D_{\mathcal C}\times E$ . Let $Y:=\pi _{D_{\mathcal C}} (f"Z)$ . Let us split to two cases:

$\blacktriangleright $ Suppose $|Y|< \lambda ^+$ . By the pigeonhole principle, we can find a subset $Q\subseteq Z$ of size $\theta $ such that $f"Q=\{c\}\times \{x\}\times E$ for some $x\in D_{\mathcal C}$ . As f is Tukey and Q is unbounded, we must have that $f"Q$ is unbounded, but this is absurd as $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}} (E))>\theta $ .

$\blacktriangleright $ Suppose $|Y|=\lambda ^+$ . As f is Tukey and either $\operatorname {\mathrm {non}}(\mathcal I_{\operatorname {\mathrm {bd}}} (E))>\theta $ or $E=1$ , every subset of Y of size $\theta $ is unbounded which is impossible by Claim 5.11.3.1.

5.4 Structure of $D_{\mathcal C}$

In [12, Lemmas 1, 2], Todorčević defined for every $\kappa $ regular and $S\subseteq \kappa $ the directed set $D(S):=\{C\subseteq [S]^{\leq \omega } \mid \forall \alpha <\omega _1[\sup (C\cap \alpha )\in C]\}$ ordered by inclusion; and studied the structure of such directed sets. In this section we follow this line of study but for directed sets of the form $D_{\mathcal C}$ , constructing a large $<_T$ -antichain and chain of directed sets using $\theta $ -support product.

5.4.1 Antichain

Theorem 5.12. Suppose $2^\lambda =\lambda ^+$ , $\lambda ^\theta <\lambda ^+$ , then there exists a family $\mathcal F$ of size $2^{\lambda ^+}$ of directed sets of the form $D_{\mathcal C}$ such that every two of them are Tukey incomparable.

Proof As $2^\lambda =\lambda ^+$ holds, by Shelah’s Theorem we get that $\diamondsuit (S)$ holds for every $S\subseteq E^{\lambda ^+}_\theta $ stationary subset. Let us fix some stationary subset $S\subseteq E^{\lambda ^+}_\theta $ and a partition of S into $\lambda ^+$ -many stationary subsets $\langle S_\alpha \mid \alpha <\lambda ^+ \rangle $ . For each $S_\alpha $ we fix a $\clubsuit (S_\alpha )$ sequence $\langle C_\beta \mid \beta \in S_\alpha \rangle $ .

Let us fix a family $\mathcal F$ of size $2^{\lambda ^+}$ of subsets of $S $ such that for every two $R,T\in \mathcal F$ there exists some $S_\alpha $ such that $R\setminus T \supseteq S_\alpha $ . For each $T\in \mathcal F$ let us define a C-sequence $\mathcal C_T:= \langle C_\alpha \mid \alpha \in T \rangle $ . Clearly the following lemma shows the family $\{ D_{\mathcal C_T} \mid T\in \mathcal F \}$ is as sought.

Claim 5.12.1. Suppose $\mathcal C_T:=\langle C_\beta \mid \beta \in T \rangle $ and $\mathcal C_R:=\langle C_\beta \mid \beta \in R \rangle $ are two C-sequences such that $T,R\subseteq E^{\lambda ^+}_\theta $ are stationary subsets. Then if $\langle C_\beta \mid \beta \in T\setminus R \rangle $ is a $\clubsuit $ -sequence, then $D_{\mathcal C_T}\not \leq _T D_{\mathcal C_R}$ .

Proof Suppose $f:D_{\mathcal C_T} \rightarrow D_{\mathcal C_R}$ is a Tukey function. Fix a subset $W\subseteq [\lambda ^+]^1\subseteq D_{\mathcal C_T}$ of size $\lambda ^+$ , we split to two cases:

$\blacktriangleright $ Suppose $f"W \subseteq [\alpha ]^{\theta }$ for some $\alpha <\lambda ^+$ . As $\lambda ^\theta <\lambda ^+$ , by the pigeonhole principle we can find a subset $X\subseteq W $ of size $\lambda ^+$ such that $f"X = \{z\}$ for some $z\in D_{\mathcal C_R}$ . As $\langle C_\beta \mid \beta \in T\setminus R \rangle $ is a $\clubsuit $ -sequence and $\bigcup X \in [\lambda ^+]^{\lambda ^+}$ , there exists some $\beta \in T\setminus R$ such that $C_\beta \subseteq \bigcup X$ . So X is an unbounded subset of $\mathcal C_T$ such that $f"X$ is bounded in $\mathcal C_R$ which is absurd.

$\blacktriangleright $ As $|f"W|=\lambda ^+$ , using $\lambda ^\theta <\lambda ^+$ we may fix a subset $Y=\{y_\beta \mid \beta <\lambda ^+\}\subseteq f"W$ which forms a $\Delta $ -system with a root $R_1$ . In other words, for $\alpha <\beta <\lambda ^+$ we have $y_\alpha \setminus R_1 < y_\beta \setminus R_1 $ and $y_\alpha \cap y_\beta =R_1$ . For each $\alpha <\lambda ^+$ , we fix $x_\alpha \in W$ such that $f(x_\alpha )=y_\alpha $ . Finally, without loss of generality we may use the $\Delta $ -system lemma again and refine our set Y to get that there exists a club $E\subseteq \lambda ^+$ such that, for all $\alpha <\beta <\lambda ^+$ we have:

• • $x_\alpha \cap x_\beta = \emptyset $ ;

• • $y_\alpha \cap y_\beta = R_1 $ ;

• • there exists some $\gamma \in E$ such that $x_\alpha < \gamma < x_\beta $ and $y_\alpha \setminus R_1 < \gamma < y_\beta \setminus R_1 $ ;

• • $f(x_\alpha ) = y_\alpha $ .

Furthermore, we may assume that between any two elements of $\xi <\eta $ in E there exists a unique $\alpha <\lambda ^+$ such that $\xi < x_\alpha \cup (y_\alpha \setminus R_1)<\eta $ .

As $\langle C_\beta \mid \beta \in T\setminus R \rangle $ is a $\clubsuit $ -sequence, there exists some $\beta \in (T\setminus R)\cap \operatorname {\mathrm {acc}}(E)$ such that $C_\beta \subseteq \bigcup \{x_\alpha \mid \alpha <\lambda ^+\}$ . Construct by recursion an increasing sequence $\langle \beta _\nu \mid \nu <\theta \rangle \subseteq C_\beta $ and a sequence $\langle z_\nu \mid \nu <\theta \rangle \subseteq \{x_\alpha \mid \alpha <\lambda ^+\}$ such that $\beta _\nu \in z_\nu <\beta $ .

Clearly, $\{z_\nu \mid \nu <\theta \}$ is unbounded in $ D_{\mathcal C_T}$ , so the following claim proves f is not a Tukey function.

Subclaim 5.12.1.1. The subset $\{ f(z_\nu )\mid \nu <\theta \}$ is bounded in $D_{\mathcal C_R}$ .

Proof Let $Y:=\bigcup f(z_\nu )$ and $\mathcal C_R:=\langle C_\beta \mid \beta \in R \rangle $ , we will show that for every $\alpha \in R$ , we have $|Y\cap C_\alpha |<\theta $ . By the refinement we did previously it is clear that $\{f(z_\nu )\setminus R_1\mid \nu <\theta \}$ is a pairwise disjoint sequence, where for each $\nu <\theta $ we have some element $\gamma _\nu \in E$ such that $f(z_\nu )\setminus R_1 < \gamma _\nu < f(z_{\nu +1}) \setminus R_1<\beta $ . Let $\alpha \in R$ .

$\blacktriangleright $ Suppose $\alpha>\beta $ . As $C_\alpha $ is cofinal in $\alpha $ and of order-type $\theta $ , then $|Y\cap C_\alpha |<\theta $ .

$\blacktriangleright $ Suppose $\alpha <\beta $ . As $C_\alpha $ is cofinal in $\alpha $ and of order-type $\theta $ , there exists some $\nu <\theta $ such that for all $\nu <\rho <\theta $ , we have $(f(z_\rho )\setminus R_1)\cap C_\alpha =\emptyset $ . As $f(z_\rho )\in D_{\mathcal C_R}$ for every $\rho <\theta $ and $\theta $ is regular, we get that $|Y\cap C_\alpha |<\theta $ as sought.

As $\beta \notin R$ there are no more cases to consider.

Corollary 5.13. Suppose $2^\lambda =\lambda ^+$ , $\lambda ^\theta <\lambda ^+$ and $S\subseteq E^{\lambda ^+}_\theta $ is a stationary subset. Then there exists a family $\mathcal F$ of directed sets of the form $D_{\mathcal C}\times [\lambda ]^{<\theta }$ of size $2^{\lambda ^+}$ such that every two of them are Tukey incomparable.

Proof Clearly by the same arguments of Theorem 5.12 the following lemma is suffices to get the wanted result.

Claim 5.13.1. Suppose $\mathcal C_T:=\langle C_\beta \mid \beta \in T \rangle $ and $\mathcal C_R:=\langle C_\beta \mid \beta \in R \rangle $ are two C-sequences such that $T,R\subseteq E^{\lambda ^+}_\theta $ are stationary subsets such that $T\setminus R$ is stationary. Then if $\langle C_\beta \mid \beta \in T\setminus R \rangle $ is a $\clubsuit $ -sequence, then $ D_{\mathcal C_T}\times [\lambda ]^{<\theta }\not \leq _T D_{\mathcal C_R}\times [\lambda ]^{<\theta }$ .

Proof Suppose $f: D_{\mathcal C_T} \times [\lambda ]^{<\theta }\rightarrow D_{\mathcal C_R}\times [\lambda ]^{<\theta }$ is a Tukey function. Consider $Q=f"([\lambda ^+]^1\times \{\emptyset \} )$ , let us split to two cases:

$\blacktriangleright $ If $|Q|<\lambda ^+$ , then by the pigeonhole principle, there exists $x\in D_{\mathcal C_R}$ , $F\in [\lambda ]^{<\theta }$ and a set $W\subseteq [\lambda ^+]^1$ of size $\lambda ^+$ such that $f"( W\times \{\emptyset \}) =\{(x,F)\}$ . As $\langle C_\beta \mid \beta \in T\setminus R \rangle $ is a $\clubsuit $ -sequence and $\bigcup W \in [\lambda ^+]^{\lambda ^+}$ , we may fix some $\beta \in T\setminus R$ such that $C_\beta \subseteq \bigcup W$ . Hence $W\times \{\emptyset \}$ is unbounded in $D_{\mathcal C_T}\times [\lambda ]^{<\theta }$ but $f"(W\times \{\emptyset \})$ is bounded in $D_{\mathcal C_R}\times [\lambda ]^{<\theta }$ which is absurd as f is Tukey.

$\blacktriangleright $ If $|Q|=\lambda ^+$ , then by the pigeonhole principle there exists some $F\in [\lambda ]^{<\theta }$ and a set $W\subseteq [\lambda ^+]^1$ of size $\lambda ^+$ such that, $f"(W\times \{\emptyset \}) \subseteq D_{\mathcal C_R}\times \{F\} $ . Let $Y:=\pi _0(f"(W\times \{\emptyset \}))$ . Next, we may continue with the same proof as in Lemma 5.12.1.

5.4.2 Chain

Theorem 5.14. Suppose $2^\lambda =\lambda ^+$ , $\lambda ^\theta <\lambda ^+$ . Then there exists a family $\mathcal F=\{D_{\mathcal C_\xi } \mid \xi <\lambda ^+\}$ of Tukey incomparable directed sets of the form $D_{\mathcal C}$ such that $\langle \prod ^{\leq \theta }_{\zeta <\xi } D_{\mathcal C_\zeta } \mid \xi <\lambda ^+\rangle $ is a $<_T$ -increasing chain.

Proof As in Theorem 5.12, we fix a partition $\langle S_\zeta \mid \zeta <\lambda ^+\rangle $ of $E^{\lambda ^+}_\theta $ to stationary subsets such that there exists a $\clubsuit (S_\zeta ) $ -sequence $\mathcal C_\zeta $ for $\zeta <\lambda ^+$ . Note that for every $A\in [\lambda ^+]^{<\lambda ^+}$ , we have $|\prod ^{\leq \theta }_{\zeta \in A} D_{\mathcal C_\zeta }|=\lambda ^+$ . Note that for every $A,B \in [\lambda ^+]^{<\lambda ^+}$ such that $A\subset B$ , we have $\prod ^{\leq \theta }_{\zeta \in A} D_{\mathcal C_\zeta } \leq _T \prod ^{\leq \theta }_{\zeta \in B} D_{\mathcal C_\zeta }$ . The following claim gives us the wanted result.

Claim 5.14.1. Suppose $A\in [\lambda ^+]^{<\lambda ^+}$ and $\xi \in \lambda ^+\setminus A$ , then $ D_{\mathcal C_\xi } \not \leq _T \prod ^{\leq \theta }_{\zeta \in A} D_{\mathcal C_\zeta } $ . In particular, $\prod ^{\leq \theta }_{\zeta \in A} D_{\mathcal C_\zeta } <_T \prod ^{\leq \theta }_{\zeta \in A} D_{\mathcal C_\zeta }\times D_{\mathcal C_\xi }$ .

Proof Let $D:=D_{\mathcal C_\xi }$ and $E:=\prod ^{\leq \theta }_{\zeta \in A} D_{\mathcal C_\zeta }$ . Note that as $2^\lambda =\lambda ^+$ , then $(\lambda ^+)^\lambda = \lambda ^+$ , so $|E|=\lambda ^+$ . Suppose $f:D \rightarrow E$ is a Tukey function. Consider $Q=f"[\lambda ^+]^1$ , let us split to cases:

$\blacktriangleright $ Suppose $|Q|< \lambda ^+$ , then by pigeonhole principle, there exists $e\in E$ and a subset $X\subseteq D$ of size $\lambda ^+$ such that $f"X =\{e\}$ . As $\langle C_\beta \mid \beta \in S_\xi \rangle $ is a $\clubsuit $ -sequence, there exists some $\beta \in S_\xi $ such that $C_\beta \subseteq \bigcup X$ . So X is an unbounded subset of D such that $f"X$ is bounded in E which is absurd.

$\blacktriangleright $ Suppose $|Q|=\lambda ^+$ . Let us enumerate $Q:=\{q_\alpha \mid \alpha <\lambda ^+\}$ . Recall that for every $\zeta \in A$ , $D_{\mathcal C_{\zeta }}\subseteq [\lambda ^+]^{\leq \theta }$ . Let $z_\alpha := \bigcup \{ q_\alpha (\zeta ) \times \{\zeta \} \mid \zeta \in A, ~q_\alpha (\zeta )\neq 0_{D_{\mathcal C_\zeta }}\}$ , notice that $z_\alpha \in [\lambda ^+ \times A]^{\leq \theta }$ . We fix a bijection $\phi :\lambda ^+\times A \rightarrow \lambda ^+$ .

As $\{\phi "z_\alpha \mid \alpha <\lambda ^+\}$ is a subset of $[\lambda ^+]^{\leq \theta }$ of size $\lambda ^+$ and $\lambda ^\theta <\lambda ^+$ , by the $\Delta $ -system lemma, we may refine our sequence Q and re-index such that $\{\phi "z_\alpha \mid \alpha <\lambda ^+\}$ will be a $\Delta $ -system with root $R'$ .

For each $\alpha <\lambda ^+$ and $\zeta \in A$ , let $y_{\alpha ,\zeta }:=\{ \beta <\lambda ^+ \mid \beta \in q_\alpha (\zeta )\}$ . We claim that for each $\zeta \in A$ , the set $\{y_{\alpha ,\zeta }\mid \zeta \in A\}$ is a $\Delta $ -system with root $R_{\zeta }:=\{\beta <\lambda ^+\mid (\beta ,\zeta )\in \phi ^{-1}[R'] \}$ . Let us show that whenever $\alpha <\beta <\lambda ^+$ , we have $y_{\alpha ,\zeta }\cap y_{\beta ,\zeta } = R_{\zeta } $ . Notice that $\delta \in y_{\alpha ,\zeta }\cap y_{\beta ,\zeta } \iff \delta \in q_\alpha (\zeta )\cap q_\beta (\zeta ) \iff (\delta ,\zeta )\in z_\alpha \cap z_\beta \iff \phi (\delta ,\zeta ) \in \phi "(z_\alpha \cap z_\beta )=\phi "z_\alpha \cap \phi "z_\beta = R' \iff (\delta ,\zeta )\in \phi ^{-1}R' \iff \delta \in R_{\zeta }$ . For each $\alpha <\lambda ^+$ , we fix $x_\alpha \in [\lambda ^+]^1$ such that $f(x_\alpha )=q_\alpha $ .

We use the $\Delta $ -system lemma again and refine our sequence such that there exists a club $C\subseteq \lambda ^+$ and for all $\alpha <\beta <\lambda ^+$ we have:

1. (1) for every $\zeta \in A$ , we have $y_{\alpha ,\zeta }\cap y_{\beta ,\zeta } = R_{\zeta } $ ;

2. (2) $x_\alpha \cap x_\beta = \emptyset $ ;

3. (3) there exists some $\gamma \in C$ such that $x_\alpha \cup (\bigcup _{\zeta \in A} (y_{\alpha ,\zeta }\setminus R_\zeta ))< \gamma < x_\beta \cup (\bigcup _{\zeta \in A } (y_{\beta ,\zeta }\setminus R_\zeta ) )$ .

Furthermore, we may assume that between any two elements of $\gamma <\delta $ in C there exists some $\alpha <\lambda ^+$ such that $\gamma <x_\alpha \cup (\bigcup _{\zeta \in A } (y_{\alpha ,\zeta }\setminus R_\zeta ))<\delta $ . We continue in the spirit of Claim 5.11.3.1.

As $\langle C_\beta \mid \beta \in S_\xi \rangle $ is a $\clubsuit $ -sequence, there exists some $\beta \in S_\xi \cap \operatorname {\mathrm {acc}}(C)$ such that $C_\beta \subseteq \bigcup \{x_\alpha \mid \alpha <\lambda ^+\}$ . Construct by recursion an increasing sequence $\langle \beta _\nu \mid \nu <\theta \rangle \subseteq C_\beta $ and a sequence $\langle w_\nu \mid \nu <\theta \rangle \subseteq \{x_\alpha \mid \alpha <\lambda ^+\}$ such that ${\beta _\nu \in w_\nu <\beta }$ .

Clearly, $\{w_\nu \mid \nu <\theta \}$ is unbounded in $ D_{\mathcal C_\xi }$ , so the following Claim proves f is not a Tukey function.

Subclaim 5.14.1.1. The subset $\{ f(w_\nu )\mid \nu <\theta \}$ is bounded in E.

Proof For each $\zeta \in A$ , let $W_\zeta := \bigcup _{\nu <\theta } f(w_\nu )(\zeta )$ . We will show that $W_\zeta \in D_{\mathcal C_\zeta }$ , as $|\{\zeta \in A\mid W_\zeta \neq \emptyset \}|\leq \theta $ this will imply that $\prod ^{\leq \theta }_{\zeta \in A} W_\zeta $ is well defined and an element of E. Clearly $f(w_\nu )\leq _E \prod ^{\leq \theta }_{\zeta \in A} W_\zeta $ for every $\nu <\theta $ , so the set $\{ f(w_\nu )\mid \nu <\theta \}$ is bounded in E as sought.

Let $\mathcal C_{\zeta }:=\langle C_\alpha \mid \alpha \in S_\zeta \rangle $ , we will show that for every $\alpha \in S_\zeta $ , we have $|W_\zeta \cap C_\alpha |<\theta $ . By the refinement we did previously it is clear that $\{f(w_\nu )(\zeta )\setminus R_\zeta \mid \nu <\theta \}$ is a pairwise disjoint sequence, where for each $\nu <\theta $ we have some element $\gamma _\nu \in C$ such that $f(w_\nu )(\zeta )\setminus R_\zeta < \gamma _\nu < f(w_{\nu +1})(\zeta )\setminus R_\zeta $ . Furthermore, $f(w_\nu )(\zeta )\subseteq \beta $ for every $\nu <\theta $ . Let $\alpha \in S_\zeta $ .

$\blacktriangleright $ Suppose $\alpha>\beta $ . As $C_\alpha $ if cofinal in $\alpha $ and of order-type $\theta $ , then $|W_\zeta \cap C_\alpha |<\theta $ .

$\blacktriangleright $ Suppose $\alpha <\beta $ . As $C_\alpha $ if cofinal in $\alpha $ and of order-type $\theta $ , there exists some $\nu <\theta $ such that for all $\nu <\rho <\theta $ , we have $(f(w_\rho )(\zeta )\setminus R_\zeta )\cap C_\alpha =\emptyset $ . As $f(w_\rho )(\zeta )\in D_{\mathcal C_\zeta }$ for every $\rho <\theta $ , we get that $|W_\zeta \cap C_\alpha |<\theta $ as sought.

As $\beta \notin S_\zeta $ there are no more cases to consider.

6 Concluding remarks

A natural continuation of this line of research is analysing the class $\mathcal D_{\kappa }$ for cardinals $\kappa \geq \aleph _\omega $ . As a preliminary finding we notice that the poset $(\mathcal P(\omega ),\subset )$ can be embedded by a function $\mathfrak F$ into the class $\mathcal D_{\aleph _\omega }$ under the Tukey order. Furthermore, for every two successive elements $A,B$ in the poset $(\mathcal P(\omega ),\subset )$ , i.e., $A\subset B $ and $|B\setminus A|=1$ , there is no directed set D such that $\mathfrak F(A)<_T D <_T \mathfrak F(B)$ . The embedding is defined via $\mathfrak F(A):= \prod ^{<\omega }_{n\in A} \omega _{n+1}$ , and the furthermore part can be proved by Lemma 4.3. As a corollary, we get that in $\operatorname {\mathrm {ZFC}}$ the cardinality of $\mathcal D_{\aleph _\omega }$ is at least $2^{\aleph _0}$ .