Fact 3.5. Let $p\in \mathbb {P}^h_\eta $ and assume that a guardrail g satisfies $g{\mathpunct {\upharpoonright }}\operatorname {\mathrm {dom}}(p)=h{\mathpunct {\upharpoonright }}\operatorname {\mathrm {dom}}(p)$ . Then, $p\in \mathbb {P}^g_\eta $ .

Proof. Induction on $\eta $ .

Proof. It is enough to show that $\Vdash _{N,\mathbb {P}}\text {"}D_0^\prime \cup \dot {D}_1$ has SFIP”. (SFIP is short for Strong Finite Intersection Property and means “every finite subset has infinite intersection”.) We show that for any $A\in D_0^\prime $ and any $\Vdash _{M,\mathbb {P}}\dot {B}\in \dot {D}_1$ , $\Vdash _{N,\mathbb {P}}A\cap \dot {B}\neq \emptyset $ . Let $p\in \mathbb {P}$ be arbitrary and . Since $p\Vdash _{M,\mathbb {P}}\dot {B}\subseteq B^\prime $ and $\Vdash _{M,\mathbb {P}}D_0\subseteq \dot {D}_1$ , we obtain $B^\prime \in D_0$ . Hence in N, we can find $c\in A\cap B^\prime $ . Let $q\leq p$ be a witness of $c\in B^\prime $ . Note that an N-generic filter G is trivially M-generic as well, so $q\Vdash _{M,\mathbb {P}}c\in \dot {B}$ implies $q\Vdash _{N,\mathbb {P}}c\in \dot {B}$ . Thus, $q\Vdash _{N,\mathbb {P}}c\in A\cap \dot {B}$ and since p is arbitrary, we have $\Vdash _{N,\mathbb {P}}A\cap \dot {B}\neq \emptyset $ .

Proof. If $\gamma \in S^-$ , any $\dot {D}^h_{\gamma +1}$ extending $\dot {D}^h_\gamma $ for $h\in H$ satisfies (3.2) since every ultrafilter contains $\omega $ . Thus, we may assume $\gamma \in S^+$ . By Definition 3.2, for each $h\in H$ we can find a $\mathbb {P}^-_\gamma *\dot {\mathbb {Q}}_\gamma $ -name $\dot {D}^\prime $ of an ultrafilter extending $(\dot {D}^h_\gamma )^-$ such that for any $\bar {q}=\langle \dot {q}_m:m<\omega \rangle $ satisfying $\Vdash _{\mathbb {P}^-_\gamma }\dot {q}_m\in \dot {Q}_{\gamma ,h(\gamma )}$ for all $m<\omega $ :

(3.5) $$ \begin{align} \Vdash_{\mathbb{P}^-_\gamma}\textstyle{\lim^{(\dot{D}^h_\gamma)^-}}\bar{q}\Vdash_{\dot{Q}_\gamma}\{m<\omega:\dot{q}_m\in \dot{H}_\gamma\} \in\dot{D}^\prime. \end{align} $$

Since $(\dot {D}^h_\gamma )^-$ is extended to $\dot {D}^h_\gamma $ and $\dot {D}^\prime $ , we can find a $\mathbb {P}_\gamma *\dot {\mathbb {Q}}_\gamma =\mathbb {P}_{\gamma +1}$ -name $\dot {D}^h_{\gamma +1}$ of an ultrafilter extending $\dot {D}^h_\gamma $ and $\dot {D}^\prime $ by Lemma 3.10. This $\dot {D}^h_{\gamma +1}$ satisfies (3.2) and we are done.

Proof. Since $\mathbb {P}$ is ccc and N is $\sigma $ -closed, N contains all maximal antichains in $\mathbb {P}^-$ and hence $\mathbb {P}^-\lessdot \mathbb {P}$ by elementarity. Moreover, we may identify a (nice) $\mathbb {P}^-$ -name of a real and a (nice) $\mathbb {P}$ -name of a real in N. Define a $\mathbb {P}^-$ -name $\tau $ by $(\sigma ,p)\in \tau :\Leftrightarrow \sigma $ is a nice $\mathbb {P}^-$ -name of a real and $p\in \mathbb {P}^-$ satisfies $p\Vdash _{\mathbb {P}}\sigma \in \dot {D}$ . We obtain $\Vdash _{\mathbb {P}}\tau =\dot {D}\cap V^{\mathbb {P}^-}$ and we are done.

Proof. For $\operatorname {\mathrm {cf}}(\gamma )>\omega $ there is nothing to do, so we assume $\operatorname {\mathrm {cf}}(\gamma )=\omega $ . Let $h\in H$ be arbitrary and S be the collection of $\bar {q}=\langle \dot {q}_m:m<\omega \rangle $ such that for some increasing $\langle \xi _m<\gamma :m<\omega \rangle $ , $\Vdash _{\mathbb {P}^-_{\xi _m}}\dot {q}_m\in \dot {Q}_{\xi _m,h(\xi _m)}$ holds for each $m<\omega $ (Note that $\xi _m\to \gamma $ since $\xi _m$ are increasing and $\operatorname {\mathrm {cf}}(\gamma )=\omega $ ). For $\bar {q}\in S$ , let . We will show:

(3.6) $$ \begin{align} \Vdash_{\mathbb{P}_\gamma}\text{"}\bigcup_{\xi<\gamma}\dot{D}^h_\xi\cup\{\dot{A}(\bar{q}):\bar{q}\in S\}\text{ has SFIP}\text{"}. \end{align} $$

If not, there exist $p\in \mathbb {P}_\gamma $ , $\xi <\gamma $ , $\mathbb {P}_\xi $ -name $\dot {A}$ of an element of $\dot {D}^h_\xi $ , $\{\bar {q}^i=\langle \dot {q}^i_m:m<\omega \rangle :i<n\}\in [S]^{<\omega }$ and increasing ordinals $\langle \xi _m^i<\gamma :m<\omega \rangle $ for $i<n$ such that $\Vdash _{\mathbb {P}^-_{\xi ^i_m}}\dot {q}_m^i\in \dot {Q}_{\xi ^i_m,h(\xi ^i_m)}$ holds for $m<\omega $ and $i<n$ and the following holds:

(3.7) $$ \begin{align} p\Vdash_{\mathbb{P}_\gamma}\dot{A}\cap \bigcap_{i<n} \dot{A}(\bar{q}^i)=\emptyset. \end{align} $$

We may assume that $p\in \mathbb {P}_\xi $ . Since all $\langle \xi ^i_m<\gamma :m<\omega \rangle $ are increasing and converge to $\gamma $ , there is $m_0<\omega $ such that $\xi _m^i>\xi $ for any $m>m_0$ and $i<n$ . By Induction Hypothesis, $p\Vdash _{\mathbb {P}_\xi }\text {"}\dot {D}^h_\xi $ is an ultrafilter” and hence we can pick $q\leq _{\mathbb {P}_\xi } p$ and $m>m_0$ such that $q\Vdash _{\mathbb {P}_\xi }m\in \dot {A}$ . Let us reorder $\{\xi _m^i:i<n\}=\{\xi ^0<\cdots <\xi ^{l-1}\}$ . Inducting on $j<l$ , we construct $q_j\in \mathbb {P}_{\xi _j}$ . Let and $j<l$ and assume we have constructed $q_{j-1}$ . Let . Since $\mathbb {P}_{\xi ^j}$ forces that all $\dot {q}^i_m$ for $i\in I_j$ are in the same centered component $\dot {Q}_{\xi ^j,h(\xi ^j)}$ , we can pick $p_j\leq q_{j-1}$ in $\mathbb {P}_{\xi ^j}$ and a $\mathbb {P}_{\xi ^j}$ -name $\dot {q}_j$ of a condition in $\dot {\mathbb {Q}}_{\xi ^j}$ such that for each $i\in I_j$ , $p_j\Vdash _{\mathbb {P}_{\xi ^j}}\dot {q}_j\leq \dot {q}^i_m$ . Let . By construction, satisfies $q^\prime \leq q\leq p$ and $q^\prime {\mathpunct {\upharpoonright }}\xi _m^i\Vdash _{\mathbb {P}_{\xi _m}} q^\prime (\xi _m^i)\leq \dot {q}^i_m$ for all $i<n$ , so in particular, $q^\prime \Vdash _{\mathbb {P}_\gamma }m\in \dot {A}\cap \bigcap _{i<n} \dot {A}(\bar {q}^i)$ , which contradicts (3.7).

Proof. Almost direct from $\Delta $ -System Lemma. (see e.g., [Reference Uribe-Zapata27, Theorem 4.3.20]).

Proof. Induct on $\gamma $ .

$\textit {Successor step.}$ Let $\bar {p}=\langle (p_m,\dot {q}_m):m<\omega \rangle \in (\mathbb {P}^h_{\gamma +1})^\omega $ be an h-uniform $\Delta $ -system with root $\nabla $ . To avoid triviality, we may assume that $\gamma \in \nabla $ . Also we may assume that $\Vdash _{\mathbb {P}^-_\gamma }\dot {q}_m\in \dot {Q}_{\gamma ,h(\gamma )}$ for each $m<\omega $ . Let and . By Induction Hypothesis, . By (3.2), Thus, $\lim ^h\bar {p}=p^\infty *\dot {q}^\infty \Vdash _{\mathbb {P}_{\gamma +1}}\{m<\omega :(p_m,\dot {q}_m)\in \dot {G}_\gamma *\dot {H}_\gamma \}\supseteq \dot {A}\cap \dot {B}\in \dot {D}^h_{\gamma +1}$ .

$\textit {Limit step.}$ Let $\gamma $ be limit and $\bar {p}=\langle p_m:m<\omega \rangle \in (\mathbb {P}^h_\gamma )^\omega $ be an h uniform- $\Delta $ -system. We use the same parameters as in Definition 3.14. Let and . Since $\bar {p}{\mathpunct {\upharpoonright }}\xi $ is also an h-uniform $\Delta $ -system with root $\nabla $ , . By Induction Hypothesis, . Let . Since $\langle \alpha _{n,m}:m<\omega \rangle $ is increasing for $n\in \left [n^{\prime \prime },n^\prime \right )$ , by (3.3), for $n\in \left [n^{\prime \prime },n^\prime \right )$ . Since $\operatorname {\mathrm {dom}}(p_m)=\operatorname {\mathrm {dom}}(p{\mathpunct {\upharpoonright }}\xi )\cup \{\alpha _{n,m}:n\in \left [n^{\prime \prime },n^\prime \right )\}$ , $p^\infty \Vdash _{\mathbb {P}_\gamma }\{m<\omega :p_m\in \dot {G}_\gamma \}=\dot {A}\cap \bigcap _{n\in \left [n^{\prime \prime },n^\prime \right )} \dot {B}_n\in \dot {D}^h_\gamma $ .

Proof. Let G be any generic filter containing $p^\infty $ . By Lemma 3.17, in particular, there exists $p_m\in G$ . Since $p_m$ forces $\varphi $ , $V[G]\vDash \varphi $ and recall that G is arbitrary containing $p^\infty $ .

Proof. Direct from the definition of “closed-uf-lim-linked” in Definition 3.2 and Definition 3.16.

Assumption 3.20.

1. (1) $\kappa <\lambda $ are uncountable regular cardinals and $\gamma =\lambda +\lambda $ .

2. (2) $\mathbb {P}_\gamma $ is a $\Gamma _{\mathrm {uf}}$ -iteration with $\Gamma _{\mathrm {uf}}$ -limits on H (with the same parameters as in Definition 3.4 and 3.9).

3. (3) H is complete and $|H|<\kappa $ .

4. (4) For $\xi <\lambda $ , $\Vdash _{\mathbb {P}^-_\xi }\dot {\mathbb {Q}}_\xi =\mathbb {C}$ , the Cohen forcing. Note that $\mathbb {C}$ is $\kappa $ - $\Gamma _{\mathrm {uf}}$ -linked by Example 3.3.

Proof. Let G be any generic filter containing and work in $V[G]$ . By Lemma 3.17, . Let $c_m\in {\omega }^\omega $ be a witness of (3.8) for each $m\in M$ .

If $j-1\in \dot {A}[G]$ , then $c_m(j-1)=m\leq \dot {\pi }[G](c_m{\mathpunct {\upharpoonright }} (j-1))=\dot {\pi }[G](t)$ for each $m\in M$ , which contradicts that $M\in \dot {D}^h[G]$ is infinite and $\dot {\pi }[G](t)$ is a natural number. Thus, $j-1\notin \dot {A}[G]$ and since G is arbitrary containing $p^\infty $ , we obtain (3.9). (3.10) is direct from Corollary 3.18.

Main Lemma 3.26. Consider the case $\Gamma _{\mathrm {uf}}=\Lambda ^{\mathrm {lim}}_{\mathrm {cuf}}$ . Then, $\mathbb {P}_\gamma $ forces $C_{[\lambda ]^{<\kappa }}\preceq _T \mathbf {BPR}$ , in particular, $\mathfrak {e}\leq {\mathfrak {e}^*}\leq \kappa $ .

Proof. Let be Cohen reals (as members of ${\omega }^\omega $ ) added in the first $\lambda $ stages. We shall show that $\bar {c}$ witnesses Fact 2.7(2), an equivalent condition of $C_{[\lambda ]^{<\kappa }}\preceq _T \mathbf {BPR}$ .

Assume towards contradiction that there exist a condition $p\in \mathbb {P}_\gamma $ and a $\mathbb {P}_\gamma $ -name of a predictor $\dot {\pi }=(\dot {A},\langle \dot {\pi }_k:k\in \dot {A}\rangle )$ such that $p\Vdash \dot {c}_\beta \sqsubset ^{\mathrm {bp}} \dot {\pi }\text { for } \kappa \text {-many }\dot {c}_\beta $ . In particular, for each $i<\kappa $ we can pick $p_i\leq p$ , $\beta _i<\lambda $ and $n_i<\omega $ such that $\beta _i\geq i$ and $ p_i\Vdash \dot {c}_{\beta _i}\sqsubset ^{\mathrm {bp}}_{n_i}\dot {\pi }$ . By extending and thinning, we may assume:

1. (1) $\beta _i\in \operatorname {\mathrm {dom}}(p_i)$ . (By extending $p_i$ .)

2. (2) All $ p_i$ follow a common guardrail $h\in H$ . ( $|H|<\kappa $ .)

3. (3) $\{p_i:i<\kappa \}$ forms a uniform $\Delta $ -system with root $\nabla $ . (By Lemma 3.15.)

4. (4) $\beta _i\notin \nabla $ , hence all $\beta _i$ are distinct. (Since $\beta _i\geq i$ , $\beta _i$ are eventually out of the finite set $\nabla $ .)

5. (5) All $n_i$ are equal to $ n^*$ .

6. (6) All $p_i(\beta _i)$ are the same Cohen condition $s\in \omega ^{<\omega }$ .

7. (7) $|s|=n^*$ . (By extending s or increasing $n^*$ .)

In particular, we have that:

(3.11) $$ \begin{align} \text{For each }i<\kappa, p_i\text{ forces }\dot{c}_{\beta_i}{\mathpunct{\upharpoonright}} n^*=s\text{ and }\dot{c}_{\beta_i}\sqsubset^{\mathrm{bp}}_{n^*}\dot{\pi}. \end{align} $$

Pick the first $\omega $ many $p_i$ and fix some bijection $i\colon \omega ^{<\omega }\to \omega $ . Fix any $n<\omega $ . (For simplicity, we assume $n\geq 3$ ). For each $\sigma \in \omega ^n$ , define $q_\sigma \leq p_{i(\sigma )}$ by extending the $\beta _{i(\sigma )}$ -th position $q_\sigma (\beta _{i(\sigma )}):=s^\frown \sigma $ . By (3.11), we have:

(3.12) $$ \begin{align} \text{For each }\sigma\in\omega^{<\omega}, q_\sigma\text{ forces }\dot{c}_{\beta_{i(\sigma)}}{\mathpunct{\upharpoonright}}(n^*+n)=s^\frown\sigma\text{ and }\dot{c}_{\beta_{i(\sigma)}}\sqsubset^{\mathrm{bp}}_{n^*}\dot{\pi}. \end{align} $$

Fix $\tau \in \omega ^{n-1}$ and we consider the sequence . When defining $q_\sigma $ we changed the $\beta _{i(\sigma )}$ -th position which is out of $\nabla $ , so $\{q_{\tau ^\frown m}:m<\omega \}$ forms a uniform $\Delta $ -system with root $\nabla $ , following some new countable partial guardrail $h^\prime $ . Since H is complete, $h^\prime $ is extended to some $h_\tau \in H$ . Note that

(3.13) $$ \begin{align} h_\tau{\mathpunct{\upharpoonright}}\nabla=h^\prime{\mathpunct{\upharpoonright}}\nabla=h{\mathpunct{\upharpoonright}}\nabla. \end{align} $$

Let $q^\infty _\tau :=\lim ^{h_\tau }\bar {q}_\tau $ .

By Lemma 3.19, $q^\infty _\tau $ follows $h_\tau $ and by Definition 3.16, $\operatorname {\mathrm {dom}}(q^\infty _\tau )=\nabla $ . Thus, by (3.13) and Fact 3.5, $q^\infty _\tau $ also follows h. By (3.12), each $q_{\tau ^\frown m}$ forces that:

(3.14) $$ \begin{align} \text{there exists }c\in{\omega}^\omega\text{ such that } c{\mathpunct{\upharpoonright}}(n^*+n)=s^\frown\tau^\frown m\text{ and }c\sqsubset^{\mathrm{bp}}_{n^*}\dot{\pi}. \end{align} $$

Thus, we are under the assumption of Lemma 3.24 and hence obtain:

(3.15) $$ \begin{align} q^\infty_\tau\text{ forces } n^*+n-1\notin\dot{A}\text{ and }\varphi_\tau, \end{align} $$

where:

(3.16) $$ \begin{align} \varphi_\tau:\equiv\text{"there exists }c\in{\omega}^\omega\text{ such that } c{\mathpunct{\upharpoonright}}(n^*+n-1)=s^\frown\tau\text{ and }c\sqsubset^{\mathrm{bp}}_{n^*}\dot{\pi}\text{".} \end{align} $$

Unfix $\tau $ and fix $\rho \in \omega ^{n-2}$ . We consider the sequence . Since all $q^\infty _{\rho ^\frown m}$ follow h and have domain $\nabla $ , they form a uniform $\Delta $ -system with root $\nabla $ and have a limit $q^\infty _\rho :=\lim ^h\bar {q}_\rho $ . Similarly, we have that $\operatorname {\mathrm {dom}}(q^\infty _\rho )=\nabla $ and $q^\infty _\rho $ follows h. Note that each $q^\infty _{\rho ^\frown m}$ forces $n^*+n-1\notin \dot {A}$ and $\varphi _{\rho ^\frown m}$ . Thus, we are under the assumption of Lemma 3.24 and hence obtain that $q^\infty _\rho $ forces:

• • $n^*+n-1\notin \dot {A}$ .

• • $n^*+n-2\notin \dot {A}$ .

• • $\text {there exists }c\in {\omega }^\omega \text { such that }c{\mathpunct {\upharpoonright }}(n^*+n-2)=s^\frown \rho \text { and }c\sqsubset ^{\mathrm {bp}}_{n^*}\dot {\pi }$ .

(The first item is direct from Corollary 3.18.) Continuing this way, we ultimately obtain with the following properties:

• • $\operatorname {\mathrm {dom}}(q^n)=\nabla $ and $q^n$ follows h (hence they form an h-uniform $\Delta $ -system).

• • $q^n$ forces $ \left [n^*,n^*+n\right ) \cap \dot {A}=\emptyset $ .

Finally, unfix n and let $q^\infty $ be the limit condition of the ultimate conditions $q^\infty :=\lim ^h\langle q^n:n<\omega \rangle $ . $q^\infty $ forces that for infinitely many $n<\omega $ , $\left [n^*,n^*+n\right ) \cap \dot {A}=\emptyset $ , which contradicts that $\dot {A}$ is infinite.

Proof. $(1)\Rightarrow (2)$ : Let $\dot {D}$ be the $\mathbb {P}$ -name of an ultrafilter extending D as in Definition 3.2. Then, r forces:

Thus, we have $B\in D$ since B is in the ground model.

$(2)\Rightarrow (1)$ : For $\bar {q}=\langle q_m:m<\omega \rangle \in Q^\omega $ , let . We show $\Vdash _{\mathbb {P}}\text {"} D\cup \{\dot {A}(\bar {q}):\bar {q}\in Q^\omega \cap V,~\lim ^D\bar {q}\in \dot {G}\}$ has SFIP”. If not, there exist $r\in \mathbb {P}$ , $A\in D$ , $n<\omega $ and $\bar {q}^j=\langle q_m^j:m<\omega \rangle \in Q^\omega $ satisfying $r\Vdash \lim ^D\bar {q}^j\in \dot {G}$ for $j<n$ , such that $r\Vdash A\cap \bigcap _{j<n}\dot {A}(\bar {q}^j)=\emptyset $ . We may assume that $r\leq \lim ^D\bar {q}^j$ for all $j<n$ . By $(\star )_n$ , we can find some $m\in A$ and $\tilde {r}\leq r$ extending all $q^j_m$ . Thus, $\tilde r\Vdash m\in A\cap \bigcap _{j<n}\dot {A}(\bar {q}^j)$ , which is a contradiction. Hence, we can take a name of an ultrafilter $\dot {D}^\prime $ extending $D\cup \{\dot {A}(\bar {q}):\bar {q}\in Q^\omega \cap V,~\lim ^D\bar {q}\in \dot {G}\}$ . Let $\bar {q}=\langle q_m:m<\omega \rangle \in Q^\omega $ be arbitrary. Since $\lim ^D\bar {q}\Vdash \lim ^D\bar {q}\in \dot {G}$ trivially holds, $\lim ^D\bar {q}\Vdash \dot {A}(\bar {q})=\{m<\omega :q_m\in \dot {G}\}\in \dot {D}^\prime $ is obtained and we are done.

Proof. For $s\in \omega ^{<\omega }$ and $k<\omega $ , let . We show is uf-lim-linked. Let D be any ultrafilter and define $\lim ^D\colon Q^\omega \to Q$ , $\bar {q}=\langle q_m=(s,k,\varphi _m):m<\omega \rangle \mapsto (s,k,\varphi ^\infty )$ as follows:

$$\begin{align*}j\in \varphi_\infty(i) \Leftrightarrow \{ m<\omega: j\in \varphi_m(i)\}\in D. \end{align*}$$

We check first $(\star )_1$ and then $(\star )_n$ for $n\geq 2$ ( $(\star )_0$ trivially holds).

The case $(\star )_1$ : Assume that $\bar {q}=\langle q_m:m<\omega \rangle \in Q^\omega $ and $r\leq \lim ^D\bar {q}$ . Let $r=(s^\prime ,k^\prime ,\varphi ^\prime )$ and $\lim ^D\bar {q}=(s,k,\varphi ^\infty )$ . Since $r\leq \lim ^D\bar {q}$ , $s^\prime (i)\notin \varphi _\infty (i)$ for all $i\in \operatorname {\mathrm {dom}}(s^\prime \setminus s)$ . That is, $\{m<\omega :s^\prime (i)\in \varphi _m(i)\}\notin D$ . Since D is an ultrafilter, $A^i :=\{m: s^\prime (i)\notin \varphi _m(i)\}\in D$ for $i\in \operatorname {\mathrm {dom}}(s^\prime \setminus s)$ . Let and for $m\in A$ , $s^\prime (i)\notin \varphi _m(i)$ for all $i\in \operatorname {\mathrm {dom}}(s^\prime \setminus s)$ . Thus, $q_m$ is compatible with r.

The case $(\star )_n$ : Suppose that:

• • $n\geq 2$ and $\bar {q}^j=\langle q_m^j:m<\omega \rangle \in Q^\omega $ for $j<n$ ,

• • $r\leq \lim ^D\bar {q}^j$ for all $j<n$

By $(\star )_1$ , . Let $m\in A$ and define $r=(s^\prime ,k^\prime ,\varphi ^\prime )$ and $q^j_m=(s,k,\varphi ^j_m)$ . Since $s^\prime \supseteq s$ , $k^\prime \ge k$ and r and $q_m^j$ are compatible, the condition extends r and all $q_m^j$ for $j<n$ where for each $i<\omega $ . Thus, A witnesses $(\star )_n$ holds.

Proof. Q in the previous proof is centered and the limit function $\lim ^D$ is closed in Q.

Proof. Fix $k<\omega $ , $d,\pi $ and $f^*=\{f^*_l\in \omega ^{|d|}:l<k\}$ where each $f^*_l$ is different. Let $Q\subseteq \mathbb {PR}_g$ consist of every $(d^\prime ,\pi ^\prime ,F^\prime )$ such that $d^\prime =d, \pi ^\prime =\pi $ and $F^\prime =\{f^\prime _l:l<k\})$ where $f^\prime _l{\mathpunct {\upharpoonright }}|d|=f^*_l$ for all $l<k$ . It is enough to show that Q is uf-lim-linked. Let D be any ultrafilter. For $\bar {f}=\langle f^i\in {\omega }^\omega :i<\omega \rangle $ , if $\bar {f}$ satisfies that for each $n<\omega $ , there (uniquely) exists $a_n<\omega $ such that $\{i<\omega :f^i(n)=a_n\}\in D$ , we define $\bar {f}^\infty \in {\omega }^\omega $ by $ \bar {f}^\infty (n)=a_n$ for each $n<\omega $ . Note that ${\bar {f}}^\infty $ is always defined if $g\in \left (\omega \setminus 2\right )^\omega $ .

For $\bar {q}=\langle q_m=(d,\pi ,\{f^m_l:l<k\})\in \mathbb {PR}:m<\omega \rangle \in Q^\omega $ , define $\lim ^D\bar {q}$ as follows:

• • , .

• • .

• • For $l\in B$ , let $n_l$ be the first $n<\omega $ where no $a<\omega $ satisfies: $\{m<\omega :f^m_l(n)=a\}\in D$ (hence $n_l\geq |d|$ ).

• • (if $B=\emptyset $ , ).

• • .

• • .

To see $\lim ^D\bar {q}$ is a condition, it is enough to show that any $\bar {f}_l^\infty ,\bar {f}_{l^\prime }^\infty \in F^\infty $ with $\bar {f}_l^\infty {\mathpunct {\upharpoonright }}|d^\infty |=\bar {f}_{l^\prime }^\infty {\mathpunct {\upharpoonright }}|d^\infty |$ satisfies $\bar {f}_l^\infty =\bar {f}_{l^\prime }^\infty $ . Since $f^*_l=\bar {f}_{l}^\infty {\mathpunct {\upharpoonright }}|d|=\bar {f}_{l^\prime }^\infty {\mathpunct {\upharpoonright }}|d|=f^*_{l^\prime }$ , we have $l=l^\prime $ and hence $\lim ^D\bar {q}$ is a valid condition.

Note that if $g\in \left (\omega \setminus 2\right )^\omega $ , then $A=k, B=\emptyset , n^\infty =|d|, d^\infty =d$ and hence $\lim ^D\bar {q}\in Q$ (for any D).

So, it is enough to show that $\lim ^D$ satisfies $(\star )_n$ for $n<\omega $ . We check first $(\star )_n$ for $n\geq 2$ assuming $(\star )_1$ and then $(\star )_1$ since the former is easier to show.

The case $(\star )_n$ : Assuming $(\star )_1$ , suppose that:

• • $n\geq 2$ and $\bar {q}^j=\langle q_m^j:m<\omega \rangle \in Q^\omega $ for $j<n$ ,

• • $r\leq \lim ^D\bar {q}^j$ for all $j<n$

By $(\star )_1$ , . Let $m\in A$ and put $r=(d^\prime ,\pi ^\prime ,F^\prime )$ and $q^j_m=(d,\pi ,F^j_m)$ . Then, the condition extends r and all $q_m^j$ for $j<n$ where N is large enough to satisfy that any distinct functions in $F^\prime \cup \bigcup _{i<n}F_i$ have different values before $|d^\prime |+N$ (hence $\tilde {r}$ is a condition and hereafter we use “N is large enough” in this sense).

The case $(\star )_1$ : Assume that $\bar {q}=\langle q_m:m<\omega \rangle \in Q^\omega $ and $r\leq \lim ^D\bar {q}=(d^\infty ,\pi ,F^\infty )$ . Let and .

Fix $l\in A$ and $n\in \mathrm {New}$ . By the definition of $\bar {f}_l^\infty $ , . Along with $r\leq \lim ^D\bar {q}$ , for $m\in X_0$ , $\pi ^r_n(f^m_l{\mathpunct {\upharpoonright }} n)=\bar {f}_l^\infty (n)=f^m_l(n)$ .

Unfixing l and n, we have:

(3.17)

Fix $l\in B$ and $n\in \mathrm {New}$ . Let . Since $l\in B$ , . Since $n_l<n^\infty \leq n$ , for all $m\in X_2, f^m_l{\mathpunct {\upharpoonright }} n\notin \operatorname {\mathrm {dom}}(\pi ^r_n)$ . Unfixing l and n, we have:

(3.18)

It is enough to show that for all $m\in X_1\cap X_3$ , $q_m$ and r are compatible. Fix such m and define (as a common extension of $q_m$ and r) as follows:

• • .

• • where N is large enough.

• • For all $n\in (d^r)^{-1}(\{1\}), \pi ^\prime _n\supseteq \pi ^r_n$ and for all $l\in B$ and $n\in \mathrm {New}, \pi ^\prime _n(f^m_l{\mathpunct {\upharpoonright }} n)=f^m_l(n)$ (This can be done by (3.18)).

$q^\prime \leq r$ trivially holds since $(d^\prime )^{-1}(\{1\})\setminus (d^r)^{-1}(\{1\})=\emptyset $ .

To see $q^\prime \leq q_m$ , we have to show:

(3.19) $$ \begin{align} \text{For all } l<k \text{ and } n\in(d^r)^{-1}(\{1\})\setminus (d)^{-1}(\{1\}), \pi^\prime_n(f^m_l{\mathpunct{\upharpoonright}} n)=f^m_l(n). \end{align} $$

If $l\in A$ , (3.17) implies (3.19), while if $l\in B$ , (3.19) holds by the definition of $\pi ^\prime $ .

Proof. Q in the previous proof is centered since any finitely many conditions $\{(d,\pi ,F_i)\in Q:i<n\}$ have a common extension where N is large enough.

Assumption 4.2.

1. (1) $\lambda _1<\cdots <\lambda _6$ are regular uncountable cardinals.

2. (2) $\lambda _3=\mu _3^+$ and $\lambda _4=\mu _4^+$ are successor cardinals and $\mu _3$ is regular.

3. (3) implies for all .

4. (4) $\lambda _6^{<\lambda _5}=\lambda _6$ , hence for all .

Proof. Direct from Fact 2.8, Fact 2.9 and Assumption 4.2 (4).

Assumption 4.5. $\lambda _6\leq 2^{\mu _3}$ .

Proof. Direct from Corollary 3.8, Assumption 4.2 (3), and 4.5.

Construction 4.7. We can construct a ccc finite support iteration of length $\gamma $ satisfying the following items:

1. (1) is a $\lambda _3$ - $\Lambda ^{\mathrm {lim}}_{\mathrm {uf}}$ -iteration and has $\Lambda ^{\mathrm {lim}}_{\mathrm {uf}}$ -limits on H with the following witnesses:

   • • $\langle \mathbb {P}_\xi ^-:\xi <\gamma \rangle $ , the complete subposets witnessing $\Lambda ^{\mathrm {lim}}_{\mathrm {uf}}$ -linkedness.

   • • $\bar {Q}=\langle \dot {Q}_{\xi ,\zeta }:\zeta <\theta _\xi ,\xi <\gamma \rangle $ , the $\Lambda ^{\mathrm {lim}}_{\mathrm {uf}}$ -linked components.

   • • $\bar {D}=\langle \dot {D}^h_\xi :\xi \leq \gamma ,~h\in H \rangle $ , the ultrafilters.

   • • , the trivial stages and , the non trivial stages.

2. (2) is also a $\lambda _4$ - $\Lambda ^{\mathrm {lim}}_{\mathrm {cuf}}$ -iteration and has $\Lambda ^{\mathrm {lim}}_{\mathrm {cuf}}$ -limits on $H^\prime $ with the following witnesses:

   • • $\langle \mathbb {P}_\xi ^-:\xi <\gamma \rangle $ , the (same as above) complete subposets witnessing $\Lambda ^{\mathrm {lim}}_{\mathrm {cuf}}$ -linkedness.

   • • $\bar {R}=\langle \dot {R}_{\xi ,\zeta }:\zeta <\theta _\xi ,\xi <\gamma \rangle $ , the $\Lambda ^{\mathrm {lim}}_{\mathrm {cuf}}$ -linked components.

   • • $\langle \dot {E}^{h^\prime }_\xi :\xi \leq \gamma ,~h^\prime \in H^\prime \rangle $ , the ultrafilters.

   • • , the trivial stages and , the non-trivial stages.

3. (3) For each $\xi \in \gamma \setminus S_0$ , $N_\xi \preccurlyeq H_\Theta $ is a submodel where $\Theta $ is a sufficiently large regular cardinal satisfying:

   1. (a) .

   2. (b) $N_\xi $ is $\sigma $ -closed, i.e., $(N_\xi )^\omega \subseteq N_\xi $ .

   3. (c) For any , $\eta <\gamma $ and set of (nice names of) reals A in $V^{\mathbb {P}_\eta }$ of size , there is some (above $\eta $ ) such that $A\subseteq N_\xi $ .

   4. (d) If , then $\{\dot {D}^h_\xi :h\in H \}\subseteq N_\xi $ .

   5. (e) If , then $\{\dot {D}^h_\xi :h\in H \},\{\dot {E}^{h^\prime }_\xi :h^\prime \in H^\prime \}\subseteq N_\xi $ .

4. (4) For $\xi \in S_0$ , and for $\xi \in \gamma \setminus S_0$ , (since $\mathbb {P}_\xi $ is ccc and $N_\xi $ is $\sigma $ -closed, $\mathbb {P}_\xi ^-\lessdot \mathbb {P}_\xi $ ).

5. (5) For each $\xi <\gamma $ , .

   (Here, does not denote a forcing poset in the ground model, but denotes the poset interpreted in the $\mathbb {P}^-_\xi $ -extension. Also note that holds since there are at most -many reals in the $\mathbb {P}^-_\xi $ -extension. Moreover, for $\xi \in S_0$ , $\dot {\mathbb {Q}}_\xi $ is always (forced to be) the same Cohen forcing $\mathbb {C}$ .)

6. (6) $\bar {Q}$ and $\bar {R}$ are determined in the canonical way: In the case of $\bar {Q}$ , if , split $\dot {\mathbb {Q}}_\xi $ into singletons and if , split the $\sigma $ - $\Lambda ^{\mathrm {lim}}_{\mathrm {uf}}$ -linked iterand $\dot {\mathbb {Q}}_\xi $ into the $\omega $ -many $\Lambda ^{\mathrm {lim}}_{\mathrm {uf}}$ -linked components. In the case of $\bar {R}$ , do it similarly.

Proof. Fact 2.7 and Construction 4.7 (3)(3c) imply for all .

For and $5$ , applying Corollary 2.16 with , we obtain .

For , Theorem 3.21 implies $C_{[\lambda _6]^{<\lambda _3}}\preceq _T\mathbf {R}_3$ .

For , Main Lemma 3.26 implies $C_{[\lambda _6]^{<\lambda _4}}\preceq _T\mathbf {R}_4^*\preceq _T\mathbf {R}_4$ .

Construction 4.9. We shall construct a fsi and submodels $\langle N_\xi :\xi \in \gamma \setminus S_0\rangle $ such that for $\xi \in S_0$ and for $\xi \in \gamma \setminus S_0$ :

1. (1) .

2. (2) $N_\xi $ is $\sigma $ -closed.

3. (3) For any , $\eta <\gamma $ and set of (nice names of) reals A in $V^{\mathbb {P}_\eta }$ of size , there is some (above $\eta $ ) such that $A\subseteq N_\xi $ .

4. (4) where .