Proof. (1) $\Rightarrow $ (2). Assume (1) holds. Suppose $\Vdash _{\mathbb {Q}_\gamma }$ “ $\langle {\dot z_n \mid n<\omega } \rangle $ is an $\omega $ -sequence such that each $\dot z_n \in \mathrm {Ult}(\check V; \dot G)$ .” We define $D \subseteq \mathbb {Q}_\gamma $ as follows:

$$ \begin{align*}\begin{matrix} D=\{p \in \mathbb{Q}_\gamma \mid \exists\text{a function }H \in V\text{ such that } \mathop{\mathrm{dom}}(H) =p\text{ and }\\p \Vdash_{\mathbb{Q}_\gamma} [\check H]=\langle {\dot z_n \mid n<\omega} \rangle\}. \end{matrix}\end{align*} $$

By showing D is dense in $\mathbb {Q}_\gamma $ , we prove that $\mathbb {Q}_\gamma $ is presaturated. Suppose $r \in \mathbb {Q}_\gamma $ . We will find $p^* \in D$ satisfying $p^* \le _{\mathbb {Q}_\gamma } r$ .

For each $n<\omega $ , let $A_n$ be a maximal antichain in $\mathbb {Q}_\gamma $ satisfying the following condition:

$$ \begin{align*} \forall p \in A_n \exists F^p_n:p \to V\text{ such that }p\Vdash_{\mathbb{Q}_\gamma}\text{"}[\check F_n^p]=\dot z_n\text{."} \end{align*} $$

Let $\dot f$ be a $\mathbb {Q}_\gamma $ -name for the function $f:\omega \to \gamma $ such that

$$\begin{align*}f(n)=\min\{\alpha<\gamma \mid A_n \cap V_\alpha \cap G\neq \emptyset\}. \end{align*}$$

Then by (1), there is some $p_0 \le _{\mathbb {Q}_\gamma } r$ and $\alpha _0<\gamma $ such that $p_0 \Vdash _{\mathbb {Q}_\gamma }$ “ $\mathop {\mathrm {range}}(\dot f) \subseteq \alpha _0$ .” We now prove the next claim.

For each $n<\omega $ , we fix a club subset $D_{p,q}^n$ of $[V_{\alpha _0}]^\omega $ for every $p,q \in A_n \cap V_{\alpha _0}$ satisfying the following condition: For $p,q \in A_n \cap V_{\alpha _0}$ with $p \neq q$ , $\{x \in [V_{\alpha _0}]^\omega \mid x \cap \bigcup p \in p\} \cap \{x \in [V_{\alpha _0}]^\omega \mid x \cap \bigcup q \in q\} \cap D_{p,q}^n=\emptyset $ . If $p=q$ , then we let $D_{p,q}^n=[V_{\alpha _0}]^\omega $ . Let $\Delta D^n$ denote the set

$$\begin{align*}\{x \in [V_{\alpha_0}]^\omega \mid \forall p,q \in x \cap A_n \cap V_{\alpha_0}(x \in D_{p,q}^n)\}. \end{align*}$$

It is easy to see that $\Delta D^n$ is club in $[V_{\alpha _0}]^\omega $ .

Let $p^*\equiv \{x \in [V_{\alpha _0}]^\omega \mid x \cap \bigcup p_0 \in p_0\} \cap \bigcap _{n<\omega } \Delta D^n$ . So we have $p^* \le _{\mathbb {Q}_\gamma } p_0 \le _{\mathbb {Q}_\gamma } r$ . We will show that $p^* \in D$ implying D is dense in $\mathbb {Q}_\gamma $ . Note that, by the definition of $p^*$ , if $x \in p^*$ , then we cannot have $p,q \in x \cap A_n \cap V_{\alpha _0}$ with $p \neq q$ satisfying $x \cap \bigcup p \in p$ and $x \cap \bigcup q \in q$ . Using this property, for each $n \in \omega $ , we define $h_n \colon p^* \to V$ as follows:

$$ \begin{align*} h_n(x)=\begin{cases} F^p_n(x \cap \bigcup p) & \text{if }\exists p\in x \cap A_n \cap V_{\alpha_0}(x \cap \bigcup p \in p),\\ \emptyset & \text{otherwise.} \end{cases} \end{align*} $$

Now define $H:p^* \to V $ by $H(x)=\langle {h_n(x) \mid n <\omega } \rangle $ . By Claim 2.3, we have $p^* \Vdash _{\mathbb {Q}_\gamma }$ “ $[\check H]=\langle {\dot z_n \mid n<\omega } \rangle $ .” Hence $p^* \in D$ , completing the proof of D being dense in $\mathbb {Q}_\gamma $ .

Now we prove (2) $\Rightarrow $ (1). Suppose that $\mathbb {Q}_\gamma $ is presaturated. From $\not \Vdash _{\mathbb {Q}_\gamma }$ “ $\mathop {\mathrm {cf}}(\gamma )>\omega $ ,” we will derive a contradiction. Assume that there exists $p \in \mathbb {Q}_\gamma $ such that $p \Vdash _{\mathbb {Q}_\gamma }$ “ $\mathop {\mathrm {cf}}(\gamma )=\omega $ .” Let G be a $\mathbb {Q}_\gamma $ -generic filter with $p \in G$ and work in $V[G]$ . Since $\mathbb {Q}_\gamma $ is presaturated, it is precipitous. Let $j:V \to M$ be the corresponding generic elementary embedding and $\langle {\delta _n \mid n<\omega } \rangle $ be a sequence of ordinals cofinal in $\gamma $ . For each $n<\omega $ , let $g_n$ be the identity function on $[V_{\delta _n}]^\omega $ . Therefore $[g_n]_G=j"V_{\delta _n}$ and $M \models $ “ $\left \vert {j"V_{\delta _n}} \right \vert =\omega $ .” Now, by presaturation of $\mathbb {Q}_\gamma $ , we know $\langle {j"V_{\delta _n} \mid n<\omega } \rangle \in M$ . Therefore $j"V_\gamma \in M$ and $M \models $ “ $\left \vert {j"V_\gamma } \right \vert =\omega $ .” Note that $M \models j"V_\gamma \in j([V_\gamma ]^\omega )$ . Since $j"V_\gamma \in M$ , there exists a function $F \in V$ such that $\mathop {\mathrm {dom}}(F) \in \mathbb {Q}_\gamma $ and $[F]_G=j"V_\gamma $ . Without loss of generality, we can assume $\mathop {\mathrm {range}}(F) \subseteq ([V_\gamma ]^\omega )^V$ . From $\mathop {\mathrm {dom}}(F) \in V_\gamma $ , we know $\left \vert {\mathop {\mathrm {dom}}(F)} \right \vert {}^V<\left \vert {V_\gamma } \right \vert {}^V$ . Therefore ${V_\gamma \setminus \bigcup \mathop {\mathrm {range}}(F) \neq \emptyset }$ . Pick an element $a^* \in V_\gamma \setminus \bigcup \mathop {\mathrm {range}}(F)$ . Then $a^* \notin F(x)$ for every $x \in \mathop {\mathrm {dom}}(F)$ . Hence $j(a^*) \notin [F]_G$ contradicting $[F]=j"V_\gamma $ .

Proof of Lemma 3.7.

We assume that $\mathop {\mathrm {RP}}(\kappa , 2^\kappa )$ and $[H_{\kappa ^+}]^\omega \setminus \mathop {\mathrm {sp}}(D)$ is stationary in $[H_{\kappa ^+}]^\omega $ for some dense $D \subseteq \mathbb {Q}_\gamma $ and derive a contradiction. Since $\{X \in [H_{\kappa ^+}]^\omega \mid X \preccurlyeq \langle {H_{\kappa ^+}, \in , \Delta } \rangle \}$ is a club subset of $[H_{\kappa ^+}]^\omega $ , by applying the Fodor-like theorem due to Jech [Reference Jech4], there exists some $p^* \in \mathbb {Q}_\gamma $ such that the following set S is stationary in $[H_{\kappa ^+}]^\omega $ : $S\equiv \{X \in [H_{\kappa ^+}]^\omega \mid X \preccurlyeq \langle {H_{\kappa ^+}, \in , \Delta } \rangle $ , $X \cap \bigcup p^* \in p^*$ , $p^*, D \in X$ , and $\neg \exists Y \in [H_{\kappa ^+}]^\omega (Y \preccurlyeq \langle {H_{\kappa ^+}, \in , \Delta } \rangle \land X \subseteq Y \land \exists q \in Y \cap D(q \le _{\mathbb {Q}_\gamma } p^* \land Y \cap \bigcup q \in q \land X \cap \bigcup p^*=Y \cap \bigcup p^*))\}$ .

Let $\alpha <\gamma $ be large enough that $p^*, \bigcup p^* \in V_\alpha $ . By $\mathop {\mathrm {RP}}(\kappa , 2^\kappa )$ , there exists $A \in [H_{\kappa ^+}]^{<\kappa }$ such that $A \preccurlyeq H_{\kappa ^+}$ , $V_\alpha \subseteq A$ , and $S \cap [A]^\omega $ is stationary in $[A]^\omega $ .

Since $\left \vert {A} \right \vert <\kappa =\left \vert {V_\gamma } \right \vert $ , there are $\overline {A} \in V_\gamma $ and a bijection $\pi :A \to \overline {A}$ such that $V_\alpha \subseteq \overline {A}$ and $\pi \restriction V_\alpha $ is the identity map. Then $\overline {S}\equiv \{\pi "X \mid X \in S \cap [A]^\omega \}$ is stationary in $[\overline {A}]^\omega $ , implying $\overline {S} \in \mathbb {Q}_\gamma $ and $\bigcup \overline {S}=\overline {A}$ .

Since D is dense in $\mathbb {Q}_\gamma $ , there exists some $\overline {q} \in D$ such that $\overline {q} \le _{\mathbb {Q}_\gamma } \overline {S}$ . Without loss of generality, we may assume that $\overline {q} \le p^*$ or $\{X \in [\bigcup p^* \cup \bigcup \overline {q}]^\omega \mid X \cap \bigcup p^* \in p^*\land X \cap \bigcup \overline {q} \in \overline {q}\}=\emptyset $ . For convenience, we denote $\{X \in [\bigcup p \cup \bigcup q]^\omega \mid X \cap \bigcup p \in p\land X \cap \bigcup q \in q\}=\emptyset $ by $p \mathrel {\bot ^*} q$ .

Now take a countable $Y^* \subseteq H_{\kappa ^+}$ such that $Y^* \preccurlyeq \langle {H_{\kappa ^+}, \in , \Delta } \rangle $ , $Y^* \cap \bigcup \overline {q} \in \overline {q}$ , and $\pi , A, \overline {A}, S \cap [A]^\omega , p^*, \overline {q} \in Y^*$ . Since $Y^* \cap \bigcup \overline {q} \in \overline {q}$ and $\overline {q} \le _{\mathbb {Q}_\gamma } \overline {S}$ , we have $Y^* \cap \overline {A}=Y^* \cap \bigcup \overline {S} \in \overline {S}$ . So there exists some $X^* \in S \cap [A]^\omega $ such that $\pi "X^*=Y^* \cap \bigcup \overline {S}$ . Since $\pi ^{-1} \in Y^*$ , $X^* \subseteq Y^*$ . From $\bigcup p^* \in V_\alpha $ and $\pi \restriction V_\alpha =\mathrm {id} \restriction V_\alpha $ , we know $\pi "\bigcup p^*=\bigcup p^*$ . Note that $Y^* \cap \bigcup p^*=\pi "X^* \cap \pi "\bigcup p^*=\pi "(X^* \cap \bigcup p^*) =X^* \cap \bigcup p^* \in p^*$ . From $Y^* \cap \bigcup \overline {q} \in \overline {q}$ and $Y^* \cap \bigcup p^* \in p^*$ , we cannot have $p^* \mathrel {\bot }^* \overline {q}$ . So we conclude $\overline {q} \le _{\mathbb {Q}_\gamma } p^*$ .

We have $X^* \in S$ and $Y^* \in [H_{\kappa ^+}]^\omega $ such that $Y^*\preccurlyeq \langle {H_{\kappa ^+}, \in , \Delta } \rangle $ , $X^* \subseteq Y^*$ , $\overline {q} \in Y^* \cap D$ , $\overline {q} \le _{\mathbb {Q}_\gamma } p^*$ , $Y^* \cap \bigcup \overline {q} \in \overline {q}$ , and $X^* \cap \bigcup p^*=Y^* \cap \bigcup p^*$ . But this contradicts the definition of S.

Proof of Lemma 3.8.

Under the hypothesis of our lemma, we will show that Player I does not have a winning strategy in the game $\Gamma _\gamma $ .

Let $\sigma $ be a strategy for Player I in $\Gamma _\gamma $ . We will show that $\sigma $ is not a winning strategy for Player I. We now define the set $C^*$ as follows:

$$ \begin{align*}\begin{matrix} C^* \equiv \{N \in [H_{\kappa^+}]^\omega \mid N \preccurlyeq \langle {H_{\kappa^+}, \in, \Delta} \rangle \land \sigma, \mathbb{Q}_\gamma \in N \land \forall D \in \\ N(D\text{ dense in }\mathbb{Q}_\gamma \to N \in \mathop{\mathrm{sp}}(D)\}. \end{matrix}\end{align*} $$

It is easy to see that $C^*$ contains a club subset of $[H_{\kappa ^+}]^\omega $ . Hence there is some ${R^* \subseteq [H_{\kappa ^+}]^\omega }$ such that, for every $N \in [H_{\kappa ^+}]^\omega $ , we have $N \in C^*$ if $N \preccurlyeq \langle {H_{\kappa ^+}, \in , \Delta , R^*} \rangle $ . From now on let us denote the structure $\langle {H_{\kappa ^+}, \in ,\Delta , R^*} \rangle $ by $\mathcal {M}^*$ .

First let $p_0 \equiv \sigma (\emptyset )$ . Take a countable set $M_0$ such that $M_0 \preccurlyeq \mathcal {M}^*$ , $p_0 \in M_0, \mathbb {Q}_\gamma \in M_0$ , and $M_0 \cap \bigcup p_0 \in p_0$ . Then let

$$ \begin{align*} D_0 \equiv \{p \in \mathbb{Q}_\gamma \mid \exists q \le_{\mathbb{Q}_\gamma} p_0(\sigma(\langle {q} \rangle)=p) \lor p \mathrel{\bot}^* p_0\}. \end{align*} $$

Then $D_0$ is dense in $\mathbb {Q}_\gamma $ and $D_0 \in M_0$ , implying $M_0 \in \mathop {\mathrm {sp}}(D_0)$ .

Since $M_0 \in \mathop {\mathrm {sp}}(D_0)$ , $p_0 \in M_0$ , and $M_0 \cap \bigcup p_0 \in p_0$ , there exist a countable set $N \preccurlyeq \langle {H_{\kappa ^+}, \in ,\Delta } \rangle $ and $p_1 \in N \cap D_0$ such that $p_1 \le _{\mathbb {Q}_\gamma } p_0$ , $M_0 \subseteq N$ , $N \cap \bigcup p_1 \in p_1$ , and $N \cap \bigcup p_0=M_0 \cap \bigcup p_0 \in p_0$ . From $N \cap \bigcup p_1 \in p_1$ and $N \cap \bigcup p_0 \in p_0$ , we cannot have $p_1 \mathrel {\bot }^* p_0$ . Thus there is some $q \le _{\mathbb {Q}_\gamma } p_0$ with $\sigma (\langle {q} \rangle )=p_1$ . Let $q_0$ be the $\Delta $ -least element such that $q_0 \le _{\mathbb {Q}_\gamma } p_0$ and $\sigma (\langle {q_0} \rangle )=p_1$ . Note that N might not be an elementary submodel of $\mathcal {M}^*$ . Now we let $M_1\equiv $ Skolem hull of $M_0 \cup (N \cap \bigcup p_1) \cup \{p_1\}$ making $M_0 \subseteq M_1 \preccurlyeq \mathcal {M}^*$ and $\{p_0, q_0, p_1\} \subseteq M_1$ .

Note that we have $M_0, M_1 \in [H_{\kappa ^+}]^\omega $ satisfying the following conditions:

1. (1) $M_0 \preccurlyeq M_1 \preccurlyeq \mathcal {M}^*$ .

2. (2) $p_0 \in M_0$ , $\{q_0, p_1\} \subseteq M_1$ .

3. (3) $\sigma (\emptyset )=p_0 \ge _{\mathbb {Q}_\gamma } q_0 \ge _{\mathbb {Q}_\gamma } \sigma (\langle {q_0} \rangle )=p_1$ .

4. (4) $M_0 \cap \bigcup p_0 \in p_0$ , $M_1 \cap \bigcup p_1 \in p_1$ , $M_1 \cap \bigcup p_0=M_0 \cap \bigcup p_0$ .

Now we repeat the above procedure using the following dense set $D_1 \in M_1$ :

$$ \begin{align*} D_1 \equiv \{p \in \mathbb{Q}_\gamma \mid \exists q \le_{\mathbb{Q}_\gamma} p_1(\sigma(\langle {q_0,q} \rangle))=p \lor p \mathrel{\bot}^* p_1\}. \end{align*} $$

By the above argument, we obtain $M_2 \in [H_{\kappa ^+}]^\omega $ satisfying:

1. (1) $M_1 \preccurlyeq M_2 \preccurlyeq \mathcal {M}^*$ .

2. (2) $\{q_1, p_2\} \subseteq M_2$ .

3. (3) $\sigma (\emptyset )=p_0 \ge _{\mathbb {Q}_\gamma } q_0 \ge _{\mathbb {Q}_\gamma } \sigma (\langle {q_0} \rangle )=p_1 \ge _{\mathbb {Q}_\gamma } q_1 \ge _{\mathbb {Q}_\gamma } \sigma (\langle {q_0,q_1} \rangle )=p_2$ .

4. (4) $M_2 \cap \bigcup p_2 \in p_2$ , $M_2 \cap \bigcup p_1=M_1 \cap \bigcup p_1$ .

By repeating the above procedure, we build a sequence $\langle {M_n,p_n,q_n \mid n<\omega } \rangle $ satisfying the following conditions:

1. (1) $M_0 \preccurlyeq M_1 \preccurlyeq M_2 \preccurlyeq M_3 \preccurlyeq \ldots \preccurlyeq \mathcal {M}^*$ .

2. (2) $\{q_n, p_{n+1}\} \subseteq M_{n+1}$ for each $n<\omega $ .

3. (3) $\sigma (\emptyset )=p_0 \ge _{\mathbb {Q}_\gamma } q_0 \ge _{\mathbb {Q}_\gamma }\sigma (\langle {q_0} \rangle )=p_1 \ge _{\mathbb {Q}_\gamma } q_1 \ge _{\mathbb {Q}_\gamma } \sigma (\langle {q_0,q_1} \rangle )=p_2 \ge _{\mathbb {Q}_\gamma } \ldots \ge _{\mathbb {Q}_\gamma } p_n \ge _{\mathbb {Q}_\gamma } q_n \ge _{\mathbb {Q}_\gamma } \sigma (\langle {q_0,\dotsc , q_n} \rangle )=p_{n+1} \ge _{\mathbb {Q}_\gamma } q_{n+1} \ge _{\mathbb {Q}_\gamma } \ldots $ .

4. (4) For each $n \in \omega $ , $M_n \cap \bigcup p_n \in p_n$ and $M_m \cap \bigcup p_m =M_n \cap \bigcup p_m$ for every $m<n$ .

We now let $X=\bigcup _{n<\omega } M_n$ . Then $X \cap \bigcup p_n \in p_n$ for every $n<\omega $ . This means that $\sigma $ is not a winning strategy for Player I.

Proof. If $\kappa $ is a limit of supercompact cardinals, then, by Lemmas 3.3 and 3.4, $\mathop {\mathrm {RP}}(\kappa ,2^\kappa )$ holds. Since $\left \vert {V_\kappa } \right \vert =\kappa $ , by Theorem 3.5, we conclude that $\mathbb {Q}_\kappa $ is precipitous.

Proof. Let $\lambda =2^\kappa $ . Fix a stationary set $S \subseteq [\lambda ]^\omega $ and $\alpha <\kappa $ . Take a $\mathord {\mathrm {Col}}(\kappa , \lambda )$ -generic filter G over V, and work in $V[G]$ . S remains a stationary subset of $[\lambda ]^\omega $ in $V[G]$ . Fix a bijection $f\colon \lambda \to \kappa $ , and let $T=\{f"x \mid x \in S\}$ . T is a stationary subset of $[\kappa ]^\omega $ .

In $V[G]$ , since $\kappa $ is a weakly compact cardinal, there is $\delta <\kappa $ such that $f"\alpha \subseteq \delta $ and $T \cap [\delta ]^\omega $ is stationary in $[\delta ]^\omega $ . Let $A=f^{-1}"\delta $ . Then we have that $\alpha \subseteq A$ , $\left \vert {A} \right \vert <\kappa $ , and $S \cap [A]^\omega $ is stationary in $[A]^\omega $ . From the $\kappa $ -closedness of $\mathord {\mathrm {Col}}(\kappa , 2^\kappa )$ , we have $A \in V$ .

Proof. We define a $\le \kappa $ -stage Easton support iteration as follows.

For $\xi <\kappa $ , suppose $\mathbb {P}_\xi $ is defined and $\mathbb {P}_\xi $ forces that “there is no weakly compact cardinal $<\xi $ .”

Case 1: There is some condition $p \in \mathbb {P}_\xi $ that forces the following statement: “ $\xi $ is a weakly compact cardinal, and the weak compactness of $\xi $ is preserved by $\mathord {\mathrm {Col}}(\xi , 2^\xi )^{V[\dot G_\xi ]}$ ” where $\dot G_\xi $ is a canonical name for a $\mathbb {P}_\xi $ -generic filter. Then take a $\mathbb {P}_\xi $ -generic filter $G_\xi $ with $p \in G_\xi $ . Clearly $V[G_\xi ]$ is a required model.

Case 2: There is no such $p \in \mathbb {P}_\xi $ . Then let $\dot {\mathbb {Q}}_\xi $ be a $\mathbb {P}_\xi $ -name such that:

$\Vdash _{\mathbb {P}_\xi } \dot {\mathbb {Q}}_\xi = \begin {cases} \text {trivial poset} & \text {if }\xi \text { is not weakly compact,}\\ \mathord {\mathrm {Col}}(\xi , 2^{\xi })^{V[\dot G_\xi ]} & \text {if }\xi \text { is weakly compact but }\mathord {\mathrm {Col}}(\xi , 2^{\xi })^{V[\dot G_\xi ]}\\ & \text { destroys the weak compactness of }\xi. \end {cases}$

We show that this procedure has to be stopped at some stage $\xi <\kappa $ . Suppose to the contrary that $\mathbb {P}_\xi $ was defined for every $\xi <\kappa $ . Let $\mathbb {P}_\kappa $ be a direct limit of the $\mathbb {P}_\xi $ ’s. By a standard argument, we know that $\left \vert {\mathbb {P}_\kappa } \right \vert =\kappa $ , $\mathbb {P}_\kappa $ satisfies the $\kappa $ -c.c., and $\mathbb {P}_\kappa $ forces that “ $\kappa $ is inaccessible and there is no weakly compact cardinal $<\kappa $ .” In V, let $\lambda =2^\kappa $ , and take a $\lambda $ -supercompact elementary embedding $j\colon V \to M$ with the critical point $\kappa $ . $j(\mathbb {P}_\kappa )$ can be factored as $\mathbb {P}_\kappa * \dot {\mathbb {Q}}_\kappa ^M * \dot {\mathbb {P}}_{\mathrm {tail}}$ . Take a $\mathbb {P}_\kappa $ -generic filter G and we work in $V[G]$ . Note that $M[G]$ is closed under taking $\lambda $ -sequences in $V[G]$ , in particular $\lambda =(2^\kappa )^{M[G]}$ , and $\mathord {\mathrm {Col}}(\kappa , \lambda )^{V[G]}=\mathord {\mathrm {Col}}(\kappa , \lambda )^{M[G]}$ . In $M[G]$ , $\kappa $ is not weakly compact, or is weakly compact but the weak compactness is destroyed by $\mathord {\mathrm {Col}}(\kappa , \lambda )^{M[G]} (=\mathord {\mathrm {Col}}(\kappa , \lambda )^{V[G]})$ . Let $\mathbb {Q}=(\dot {\mathbb {Q}}_\kappa ^M)_G$ . By the construction of $\mathbb {P}_\kappa $ , $\mathbb {Q}$ is either a trivial poset, or $\mathord {\mathrm {Col}}(\kappa , \lambda )^{V[G]}$ . Note that $\mathbb {Q}$ is $\kappa $ -directed closed in either cases. Take a $\mathbb {Q}$ -generic filter H over $V[G]$ . We know that $\kappa $ is not weakly compact in $M[G][H]$ . We derive a contradiction by proving the following claim.

Proof. Fix a sufficiently large cardinal $\mu>\lambda $ . It enough to show that $\mathord {\mathrm {Col}}(\kappa ,\mathop <\lambda ) *\dot {\mathbb {Q}}$ forces $\mathop {\mathrm {RP}}(\lambda , \mu )$ . Let $j:V \to M$ be a $\mu $ -supercompact embedding with the critical point $\lambda $ . Let us consider a poset $\mathord {\mathrm {Col}}(\kappa , \mathop <j(\lambda ))^M$ that is $\kappa $ -closed. It is known that there is a name $\dot {\mathbb {R}}$ for a $\kappa $ -closed poset such that $\mathord {\mathrm {Col}}(\kappa , \mathop <j(\lambda ))^M \approx \mathord {\mathrm {Col}}(\kappa , \mathop < \lambda ) * \dot {\mathbb {Q}} *\dot {\mathbb {R}}$ . Now choose a $\mathord {\mathrm {Col}}(\kappa , \mathop <j(\lambda ))^M$ -generic filter $j(G)$ over V, and let $G=j(G) \cap \mathord {\mathrm {Col}}(\kappa , \mathop <\lambda )$ . G is a $\mathord {\mathrm {Col}}(\kappa , \mathop <\lambda )$ -generic filter over V. In $V[j(G)]$ , we can extend j to $j:V[G] \to M[j(G)]$ by a canonical way. Let $\mathbb {Q}=\dot {\mathbb {Q}}_G$ , and take a $\mathbb {Q}$ -generic filter H over $V[G]$ induced by $j(G)$ . We have $H \in M[j(G)]$ , and $j"H \in M[j(G)]$ . In $M[j(G)]$ , since $\mu $ was sufficiently large, the size of $j"H$ is $\le \mu $ , hence is $<j(\lambda )$ . Since $j(\mathbb {Q})$ is $j(\lambda )$ -directed closed in $M[j(G)]$ , we can find a condition $p^* \in j(\mathbb {Q})$ with $p^* \le j(q)$ for every $q \in H$ . Then take a $j(\mathbb {Q})$ -generic filter $j(H)$ over $V[j(G)]$ with $p^* \in j(H)$ , and, in $V[j(G)][j(H)]$ , we can extend j to $j:V[G][H] \to M[j(G)][j(H)]$ .

We now show that $\mathrm {RP}(\lambda , \mu )$ holds in $V[G][H]$ . Take a stationary $S \subseteq [\mu ]^\omega $ and $\alpha <\lambda $ . Since $\dot {\mathbb {R}}_G$ is $\sigma $ -closed in $V[G][H]$ and $j(\mathbb {Q})$ is $\sigma $ -closed in $V[j(G)]$ , S remains stationary in $V[j(G)][j(H)]$ , hence $j"S$ is also stationary in $[j"\mu ]^\omega $ as well. We know $\left \vert {j"\mu } \right \vert =\mu <j(\lambda )$ and $j"S = j(S) \cap [j"\mu ]^\omega \in M[j(G)][j(H)]$ . Thus,

$$ \begin{align*}\begin{matrix} M[j(G)][j(H)] \models \exists A(A \in \mathcal{P}_{j(\lambda)}j(\mu) \land j(\alpha) \subseteq A \land j(S) \cap [A]^\omega\text{ is }\\\text{stationary in }[A]^\omega). \end{matrix}\end{align*} $$

By the elementarity of j,

$$ \begin{align*} V[G][H] \models \exists A(A \in \mathcal{P}_{\lambda}\mu \land \alpha \subseteq A \land S \cap [A]^\omega\text{ is stationary in }[A]^\omega).\\[-31pt] \end{align*} $$

Proof. Let $\langle {\kappa _\xi \mid \xi \in \mathrm {ON}} \rangle $ be the increasing enumeration of all supercompact cardinals. To simplify our argument, we assume that there is no inaccessible cardinal which is a limit of supercompact cardinals: If such a cardinal exists, let $\delta $ be the least inaccessible cardinal which is a limit of supercompact cardinals, and we identify $V_\delta $ as our universe V. So we know that $\kappa _\xi>\sup _{\zeta <\xi } \kappa _\zeta $ for every $\xi $ , and if $\xi $ is a limit ordinal then $\sup _{\zeta <\xi } \kappa _\zeta $ is a singular cardinal.

We define a class iteration $\langle {\mathbb {P}_\xi , \dot {\mathbb {Q}}_\xi \mid \xi \in \mathrm {ON}} \rangle $ as follows. First, let $\mathbb {P}_0$ be a trivial poset, and $\dot {\mathbb {Q}}_0$ be a $\mathbb {P}_0$ -name for a poset $\mathord {\mathrm {Col}}(\omega _1, \mathop <\kappa _0)$ . Put $\mathbb {P}_1=\mathbb {P}_0*\dot {\mathbb {Q}}_0$ . Suppose $\mathbb {P}_\xi $ is defined such that $\left \vert {\mathbb {P}_\xi } \right \vert <\kappa _{\xi }$ . Moreover, suppose that $\mathbb {P}_{\xi }$ forces that $\kappa _{\xi -1}$ to be regular if $\xi $ is successor, and that $\kappa =\sup _{\zeta <\xi } \kappa _\zeta $ to be a cardinal if $\xi $ is limit. Let $\dot {\mathbb {Q}}_\xi $ be a $\mathbb {P}_{\xi }$ -name such that:

$$\begin{align*}\Vdash_{\mathbb{P}_\xi} \dot {\mathbb{Q}}_{\xi}=\begin{cases} \mathord{\mathrm{Col}}(\kappa_{\xi-1}, \mathop < \kappa_{\xi})^{V[\dot G_\xi]} & \text{if }\xi\text{ is successor,}\\ \mathord{\mathrm{Col}}(\kappa^+, \mathop < \kappa_{\xi})^{V[\dot G_\xi]} & \text{if }\xi\text{ is limit, where }\kappa=\sup_{\zeta<\xi} \kappa_\zeta. \end{cases} \end{align*}$$

Let $\mathbb {P}_{\xi +1}=\mathbb {P}_\xi *\dot {\mathbb {Q}}_\xi $ . If $\xi $ is limit, then let $\mathbb {P}_\xi $ be an inverse limit of the $\mathbb {P}_\zeta $ ’s for $\zeta <\xi $ . Finally, let $\mathbb {P}_{\infty }$ be a class poset that is a direct limit of the $\mathbb {P}_\xi $ ’s.

By a standard argument, we can check the following conditions hold:

1. (1) $\mathbb {P}_\infty $ forces $\mathsf {ZFC}$ .

2. (2) $\mathbb {P}_\infty $ forces “for every regular uncountable cardinal $\delta $ , the successor cardinal $\delta ^+$ is $\kappa _\xi $ for some $\xi $ .”

3. (3) For a successor $\xi $ , $\mathbb {P}_\xi $ forces that $\kappa _{\xi -1}$ to be regular, and that “ $\kappa _\xi $ is supercompact and $\dot {\mathbb {Q}}_\xi =\mathord {\mathrm {Col}}(\kappa _{\xi -1}, \mathop < \kappa _\xi )^{V[\dot G_\xi ]}$ .” Furthermore $\mathbb {P}_\infty $ can be factored as $\mathbb {P}_\xi *\mathord {\mathrm {Col}}(\kappa _{\xi -1}, \mathop <\kappa _\xi )^{V[\dot G_\xi ]}*\dot {\mathbb {P}}_{\xi , \infty }$ such that $\mathbb {P}_{\xi }*\mathord {\mathrm {Col}}(\kappa _{\xi -1},\mathop <\kappa _\xi )^{V[\dot G_\xi ]} (=\mathbb {P}_{\xi +1})$ forces “ $\dot {\mathbb {P}}_{\xi , \infty }$ is $\kappa _\xi $ -directed closed.”

4. (4) For a limit $\xi $ , let $\kappa =\sup _{\zeta <\xi } \kappa _\zeta $ . Then $\mathbb {P}_\xi $ forces that $\kappa $ to be a singular cardinal, and that “ $\kappa _\xi $ is supercompact and $\dot {\mathbb {Q}}_\xi =\mathord {\mathrm {Col}}(\kappa ^+, \mathop < \kappa _\xi )^{V[\dot G_\xi ]}$ .” Furthermore $\mathbb {P}_\infty $ can be factored as $\mathbb {P}_\xi *\mathord {\mathrm {Col}}(\kappa ^+,\mathop <\kappa _\xi )^{V[\dot G_\xi ]}*\dot {\mathbb {P}}_{\xi , \infty }$ such that $\mathbb {P}_{\xi }*\mathord {\mathrm {Col}}(\kappa ^+,\mathop <\kappa _\xi )^{V[\dot G_\xi ]} (=\mathbb {P}_{\xi +1})$ forces “ $\dot {\mathbb {P}}_{\xi , \infty }$ is $\kappa _\xi $ -directed closed.”

We show that a generic extension via $\mathbb {P}_\infty $ is as required. Take a $\mathbb {P}_\infty $ -generic filter G over V and work in $V[G]$ . Fix a regular uncountable cardinal $\delta $ . We can find $\xi $ such that $\delta ^+=\kappa _\xi $ . By (2), (3) above, and Lemma 4.7, $\mathop {\mathrm {RP}}(\delta ^+, \lambda )$ holds for every $\lambda>\delta $ .

Proof. By Lemma 4.8, we can work in a generic extension where $\mathop {\mathrm {RP}}(\delta ^+, \lambda )$ holds for every regular uncountable cardinal $\delta $ and cardinal $\lambda>\delta $ . We now take a limit ordinal $\gamma>\omega $ and let $\kappa =\left \vert {V_\gamma } \right \vert $ . $\kappa $ must be a limit cardinal. By our assumption and Lemma 3.4, $\mathop {\mathrm {RP}}(\kappa , 2^\kappa )$ holds, and, from Theorem 3.5, $\mathbb {Q}_\gamma $ is precipitous.