Definition 1.1. The conditions of the forcing ${\mathbb S^{h}_\kappa }$ are $\kappa $ -trees $T\subseteq {}^{<\kappa }\kappa $ that satisfy the following properties:

1. (i) For any $u\in T$ there exists $v\in T$ such that $u\subseteq v$ and v is splitting.

2. (ii) If $u\in \mathrm {Split}_\alpha (T)$ , then u is an $h(\alpha )$ -splitting node in T.

3. (iii) If $C\subseteq T$ is a chain of splitting nodes with $|C|<\kappa $ , then $\textstyle \bigcup C$ is a splitting node in T.

The order is defined as $T\leq _{{\mathbb S^{h}_\kappa }} S$ (here T provides more information than S) iff:

• • $T\subseteq S$ and

• • for every $u\in T$ , if $\mathrm {suc}(u,T)\neq \mathrm {suc}(u,S)$ , then $|\mathrm {suc}(u,T)|<|\mathrm {suc}(u,S)|$ .

If the forcing notion is clear from context, we will write $T\leq S$ in place of $T\leq _{{\mathbb S^{h}_\kappa }}S$ .

Proof Suppose that $C\subseteq T$ is a chain such that $u=\textstyle \bigcup C$ and $\mathrm {dom}(u)\in \kappa $ . If we show that $u\in T$ , then $\mathrm {suc}(u,T)\neq \varnothing $ by (i) of Definition 1.1, thus it follows that C is not a branch. Let

$$ \begin{align*} C'={\left\{{v\subseteq u\mid v\text{ is splitting in }T}\right\}.} \end{align*} $$

From (i) of Definition 1.1 we can conclude that either $\textstyle \bigcup C'=u$ or there is some splitting node $v\in T$ with $w\subseteq v$ for all $w\in C$ . In the first case it follows that $u\in T$ by (iii) of Definition 1.1, and in the second case $v\restriction \mathrm {dom}(u)=u\in T$ since T is a $\kappa $ -tree.

Claim 1.4. If $u\in T=\textstyle \bigcap _{\xi \in \lambda } T_\xi $ , then there is $\eta \in \lambda $ such that $\mathrm {suc}(u,T)=\mathrm {suc}(u,T_\eta )$ .

Proof Suppose that $u\in T$ , and let $\lambda _\xi =|\mathrm {suc}(u,T_{\xi })|$ , then the ordering on ${\mathbb S^{h}_\kappa }$ dictates that $\left \langle \lambda _\xi \mid \xi \in \lambda \right \rangle $ is a descending sequence of cardinals, hence there is $\eta \in \lambda $ such that $\lambda _\xi =\lambda _\eta $ for all $\xi \in [\eta ,\lambda )$ . But then $\mathrm {suc}(u,T_\xi )=\mathrm {suc}(u,T_\eta )$ for all $\xi \in [\eta ,\lambda )$ by the ordering of ${\mathbb S^{h}_\kappa }$ .

Proof of Lemma 1.3

We show that $T=\textstyle \bigcap _{\xi \in \lambda } T_\xi $ satisfies the lemma by verifying points (i)–(iii) from Definition 1.1 and showing that $T\leq T_\xi $ for all $\xi \in \lambda $ .

(i) Let $u\in T$ , and let $f\in [T]$ be a branch for which $u\subseteq f$ . If $\mathrm {ot}(f)<\kappa $ , then $f\in T_\xi $ for each $\xi $ , thus by Claim 1.4 there is some $\eta \in \lambda $ for which $\mathrm {suc}(f,T_\eta )=\mathrm {suc}(f,T)=\varnothing $ . Then clearly $T_\eta \notin {\mathbb S^{h}_\kappa }$ , which is a contradiction, hence $\mathrm {ot}(f)=\kappa $ .

Let $C_\xi ={\left \{{\alpha \in [\mathrm {ot}(u),\kappa )\mid f\restriction \alpha \text { is splitting in }T_\xi }\right \}}$ , then since $T_\xi \in {\mathbb S^{h}_\kappa }$ satisfies (i) and (iii), we see that $ C_\xi $ is a club set. But then $\textstyle \bigcap _{\xi \in \lambda } C_\xi $ is club. Any $v\in \textstyle \bigcap _{\xi \in \lambda } C_\xi $ is splitting in all $T_\xi $ , thus by Claim 1.4 it is splitting in T, and by definition of $ C_\xi $ it follows that $u\subseteq v$ .

(ii) If $u\in \mathrm {Split}_\alpha (T)$ , then by Claim 1.4 there is $\eta \in \lambda $ such that $\mathrm {suc}(u,T_\eta )=\mathrm {suc}(u,T)$ . Therefore $u\in \mathrm {Split}_\beta (T_\eta )$ for some $\beta \geq \alpha $ , hence u is a $h(\beta )$ -splitting in T. Remember that we assume h is increasing, so u is also $h(\alpha )$ -splitting in T.

(iii) Let $C\subseteq T$ be a chain of splitting nodes, then for every $\xi \in \lambda $ we also see that C is a chain of splitting nodes in $T_\xi $ , and thus $\textstyle \bigcup C$ is a splitting node in all $T_\xi $ , hence by Claim 1.4, $\textstyle \bigcup C$ is splitting in T.

( $\leq $ ) Clearly $T\subseteq T_\xi $ for each $\xi \in \lambda $ , and if $u\in T$ and $\mathrm {suc}(u,T)\neq \mathrm {suc}(u,T_\xi )$ , then by Claim 1.4 there exists $\eta \in \lambda $ such that $\mathrm {suc}(u,T)=\mathrm {suc}(u,T_\eta )$ , and clearly $\xi <\eta $ . Since $T_\xi \leq T_\eta $ by assumption, then $|\mathrm {suc}(u,T)|=|\mathrm {suc}(u,T_\eta )|<|\mathrm {suc}(u,T_\xi )|$ . Hence $T\leq T_\xi $ .

Definition 1.6. For $T,S\in {\mathbb S^{h}_\kappa }$ , we let $T\leq _\alpha S$ iff $T\leq S$ and $\mathrm {Split}_\alpha (T)=\mathrm {Split}_\alpha (S)$ . A fusion sequence is a sequence $\left \langle T_\alpha \in {\mathbb S^{h}_\kappa }\mid \alpha \in \kappa \right \rangle $ such that $T_{\beta }\leq _\alpha T_\alpha $ for all $\alpha \leq \beta \in \kappa $ .

Proof Clearly T is a $\kappa $ -tree on ${}^{<\kappa }\kappa $ . We check conditions (i)–(iii) of Definition 1.1.

(i) Let $u\in T$ and $\alpha =\mathrm {ot}(u)$ , then for any $\beta \geq \alpha $ we see that $\mathrm {Split}_\alpha (T_\beta )=\mathrm {Split}_\alpha (T_\alpha )$ . Since $\alpha =\mathrm {ot}(u)$ , necessarily there exists some $v\in \mathrm {Split}_\alpha (T_\alpha )$ such that $u\subseteq v$ , and since $v\in \mathrm {Split}_\alpha (T_\beta )$ for all $\beta>\alpha $ we see that $v\in \mathrm {Split}_\alpha (T)$ .

(ii) Let $u\in \mathrm {Split}_\alpha (T)$ , then u is $h(\alpha )$ -splitting in $T_{\alpha +1}$ . Let $\lambda _u=|\mathrm {suc}(u,T_{\alpha +1})|\geq h(\alpha )$ and let $\left \langle v_\xi \mid \xi \in \lambda _u\right \rangle $ enumerate those $v\supseteq u$ such that $v\in \mathrm {Split}_{\alpha +1}(T_{\alpha +1})$ . For all $\beta \geq \alpha +1$ we have $\mathrm {Split}_{\alpha +1}(T_{\beta })=\mathrm {Split}_{\alpha +1}(T_{\alpha +1})$ , therefore for each $\xi \in \lambda _u$ we see that $v_\xi \in T_\beta $ for all $\beta>\alpha $ , thus $v_\xi \in T$ . Therefore u is $h(\alpha )$ -splitting in T.

(iii) Let $C\subseteq T$ be a chain of splitting nodes with $|C|<\kappa $ and let $\gamma \in \kappa $ be large enough such that $C\subseteq \textstyle \bigcup _{\alpha <\gamma }\mathrm {Split}_\alpha (T)$ . It follows that $C\subseteq \textstyle \bigcup _{\alpha <\gamma }\mathrm {Split}_\alpha (T_\gamma )$ , and thus $\textstyle \bigcup C\in \mathrm {Split}_\beta (T_\gamma )$ for some $\beta \leq \gamma $ . Then also $\textstyle \bigcup C\in \mathrm {Split}_\beta (T_{\gamma '})$ for all $\gamma '>\gamma $ , hence $\textstyle \bigcup C\in \mathrm {Split}_\beta (T)$ .

( $\leq $ ) Clearly $T\subseteq T_\alpha $ for all $\alpha \in \kappa $ . Given $u\in T$ and $\alpha \in \kappa $ such that $\mathrm {suc}(u,T)\neq \mathrm {suc}(u,T_\alpha )$ , we will show that $|\mathrm {suc}(u,T)|<|\mathrm {suc}(u,T_\alpha )|$ . We may assume without loss of generality that u is splitting in T, so let $\beta \in \kappa $ be such that $u\in \mathrm {Split}_\beta (T)$ . Since $\mathrm {Split}_\gamma (T)=\mathrm {Split}_\gamma (T_\alpha )$ for all $\gamma \leq \alpha $ , we see that $\beta \geq \alpha $ . We have $\mathrm {Split}_{\beta +1}(T_{\beta +1})=\mathrm {Split}_{\beta +1}(T)$ , and thus $\mathrm {suc}(u,T_{\beta +1})=\mathrm {suc}(u,T)$ . Finally $T_{\beta +1}\leq T_\alpha $ gives us $|\mathrm {suc}(u,T)|=|\mathrm {suc}(u,T_{\beta +1})|<|\mathrm {suc}(u,T_\alpha )|$ .

Proof We will let $F:\alpha \mapsto h(\alpha )^{|\alpha |}$ and show that ${\mathbb S^{h}_\kappa }$ has the F-Sacks property.

Let $T_0\in {\mathbb S^{h}_\kappa }$ and let $\dot f$ be a name such that $T_0\Vdash \text {"}\,\dot f:\check \kappa \to \check \kappa \,\text {"}$ . If $T_0\Vdash \text {"}\,\dot f\in \check {\mathbf V}\,\text {"}$ , then the existence of an appropriate F-slalom is obvious, so we assume that $T_0\Vdash \text {"}\,\dot f\notin \check {\mathbf V}\,\text {"}$ . We will construct a fusion sequence $\left \langle T_\xi \mid \xi \in \kappa \right \rangle $ and sets ${\left \{{D_\xi \subseteq \kappa \mid \xi \in \kappa }\right \}}$ with $|D_\xi |\leq F(\xi )$ such that

$$ \begin{align*} \textstyle\bigcap_{\xi\in\kappa} T_\xi=T\Vdash\text{"}\,\dot f(\check\xi)\in \check D_\xi\,\text{"} \end{align*} $$

for each $\xi \in \kappa $ . Consequently, we can define the F-slalom $\varphi :\xi \mapsto D_\xi $ in the ground model, then it follows that $T\Vdash \text {"}\,\dot f(\check \xi )\in \check \varphi (\check \xi )\,\text {"}$ for all $\xi \in \kappa $ .

In general, we will assume each $T_\xi $ has the following property:

$$ \begin{align*} (*)\quad\text{For every }u\in\mathrm{Split}_\alpha(T_\xi)\text{ with }\alpha<\xi\text{ we have }|\mathrm{suc}(u,T_\xi)|=h(\alpha). \end{align*} $$

This is vacuously true for $T_0$ , and by using sharp $\kappa $ -trees at successor stages of our construction, $(*)$ will follow by induction. If $\gamma $ is limit, we will let $T_\gamma =\textstyle \bigcap _{\xi \in \gamma }T_\xi $ , which will be a condition by the proof of Lemma 1.3. $T_\gamma $ need not necessarily be a sharp $\kappa $ -tree, but it is at least sharp for all splitting levels less than $\gamma $ , which is enough for $(*)$ .

Suppose $T_\xi $ has been defined, then we will define $T_{\xi +1}$ such that it limits the possible values of $\dot f(\xi )$ and such that $T_{\xi +1}\leq _\xi T_\xi $ . First note that if $T_\xi $ has property $(*)$ , then $T_\xi ^*\leq _\xi T_\xi $ : If u is splitting in $T_\xi $ and $u\notin T_\xi ^*$ , then u was removed because there is some $v\subseteq u$ such that $\mathrm {suc}(v,T_\xi )$ is too large for sharpness. But then by $(*)$ it follows that $v\in \mathrm {Split}_\alpha (T_\xi )$ for some $\alpha \geq \xi $ , hence $u\in \mathrm {Split}_\beta (T_\xi )$ for some $\beta>\xi $ .

We define a set $V_\xi $ of successor nodes of the $\xi $ -th splitting level, that is,

$$ \begin{align*} V_\xi = \textstyle\bigcup\{\mathrm{suc}(u,T_\xi^*)\mid u\in\mathrm{Split}_\xi(T_\xi^*)\}. \end{align*} $$

Our goal is to find a stronger condition below each subtree $(T^*_\xi )_v$ with $v\in V_\xi $ that decides $\dot f(\check \xi )$ , and glue these conditions back together to get a condition stronger than $T^*_\xi $ . Since the size of $V_\xi $ is limited, this limits the possible values of $\dot f(\check \xi )$ to a small set.

For each $v\in V_\xi $ find a condition $T^v\leq (T_\xi ^*)_v$ such that $T^v\Vdash \text {"}\,\dot f(\check \xi )=\check \beta _\xi ^v\,\text {"}$ for some $\beta _\xi ^v\in \kappa $ . Choose some arbitrary $u\in \mathrm {Split}_\xi (T^v )$ and $w\in \mathrm {suc}(u,T^v )$ , and consider the subtree $(T^v)_{w}$ of $T^v $ generated by the initial segment w. We let $G_\xi :V_\xi \to \mathcal P(T_\xi )$ send $v\mapsto (T^v)_w$ . Note that the $\alpha $ -th splitting level of $G_\xi (v)=(T^v)_w$ corresponds to the $(\xi +1+\alpha )$ -th splitting level of $T^v$ .

Now we define

$$ \begin{align*} T_{\xi+1}&= \textstyle\bigcup G_\xi[V_\xi ]= \textstyle\bigcup {\left\{{G_\xi(v)\mid v\in V_\xi }\right\}},\\ D_{\xi}&={\left\{{\beta_\xi^v\mid v\in V_\xi }\right\}}. \end{align*} $$

For each $v\in V_\xi $ we have $v\in G_\xi (v)$ , thus each successor of a splitting node in $\mathrm {Split}_\xi (T_\xi )$ is in $T_{\xi +1}$ . Therefore we see that $\mathrm {Split}_{\xi }(T_{\xi +1})=\mathrm {Split}_\xi (T_\xi )$ . If $u\in \mathrm {Split}_{\xi +1+\alpha } (T_{\xi +1})$ for some $\alpha \in \kappa $ , then $u\in \mathrm {Split}_{\alpha }(G_\xi (v))$ , thus $u\in \mathrm {Split}_{\xi +1+\alpha }(T^v)$ , and since $T^v \in {\mathbb S^{h}_\kappa }$ , we see that u is $h(\xi +1+\alpha )$ -splitting. Therefore $T_{\xi +1}$ satisfies (ii) of Definition 1.1. It is easy to check (i) and (iii), thus we can conclude that $T_{\xi +1}\in {\mathbb S^{h}_\kappa }$ and that $T_{\xi +1}\leq _\xi T_\xi $ .

Note that the set $D_\xi $ is indeed small enough:

$$ \begin{align*} |D_\xi|\leq|V_\xi |=|\mathrm{Split}_\xi(T_{\xi}^*)|\cdot h(\xi)\leq h(\xi)^{|\xi|}=F(\xi). \end{align*} $$

For each $v\in V_\xi $ we have $T^v \Vdash \text {"}\,\dot f(\check \xi )\in \check D_\xi \,\text {"}$ , and ${\left \{{T^v \mid v\in V_\xi }\right \}}$ is predense below $T_{\xi +1}$ ; thus,

$$ \begin{align*} T_{\xi+1}\Vdash\text{"}\,\dot f(\check\xi)\in \check D_\xi\,\text{"}. \end{align*} $$

Let $T=\textstyle \bigcap _{\xi \in \kappa }T_\xi $ , then by the fusion lemma $T\in {\mathbb S^{h}_\kappa }$ , and $T\Vdash \text {"}\,\dot f(\check \xi )\in \check D_\xi \,\text {"}$ for all $\xi \in \kappa $ .

Proof Given an ${\mathbb S^{h}_\kappa }$ -name $\dot f$ and $T\in {\mathbb S^{h}_\kappa }$ such that $T\Vdash \text {"}\,\dot f:\check \kappa \to \check \kappa ^+\,\text {"}$ , then using (the proof of) the F-Sacks property we may produce sets $D_\xi $ with $|D_\xi |=F(\xi )<\kappa $ for each $\xi \in \kappa $ such that $T'\Vdash \text {"}\,\dot f(\check \xi )\in \check D_\xi \,\text {"}$ for some stronger $T'\leq T$ , and thus $\dot f$ is forced to have a range contained in $\textstyle \bigcup _{\xi \in \kappa } D_\xi $ and cannot be cofinal in $\kappa ^+$ .

Proof For $\alpha _0\in \kappa $ we can recursively define $\alpha _{n+1}$ large enough such that $\mathrm {Split}_{\alpha _n}(T)\subseteq {}^{\leq \alpha _{n+1}}\kappa $ for each $n\in \omega $ . Let $\alpha =\textstyle \bigcup _{n\in \omega }\alpha _n$ , then $\alpha \in C_T$ , hence $C_T$ is unbounded. It is easy to see that $C_T$ is continuous.

Proof Let h be such that $F(\alpha )<h(\alpha )$ for all $\alpha \in S$ , where S is a stationary subset of $\kappa $ . We will show that $\mathbb S^h_\kappa $ does not have the F-Sacks property.

Let $\dot f$ be a name for the generic $\mathbb S^h_\kappa $ -real in ${}^{\kappa }\kappa $ , let $\varphi $ be an F-slalom, let $T\in \mathbb S^h_\kappa $ and let $\alpha _0\in \kappa $ . We want to find some $\alpha \geq \alpha _0$ and $S\leq T$ such that $S\Vdash \text {"}\,\dot f(\check \alpha )\notin \check \varphi (\check \alpha )\,\text {"}$ . If we can find $u\in T\cap {}^{\alpha +1}\kappa $ such that $u(\alpha )\notin \varphi (\alpha )$ , then $(T)_u$ will be sufficient.

Let $C_T$ be as defined in Lemma 1.10 and $\alpha \in C_T\cap S$ such that $\alpha _0\leq \alpha $ , then $\mathrm {Split}_\alpha (T)=T\cap {}^\alpha \kappa $ , thus each $t\in T\cap {}^\alpha \kappa $ is an $h(\alpha )$ -splitting node. Hence, there is a set $X\subseteq \kappa $ with $|X|=h(\alpha )$ such that $t^\frown \gamma \in T$ for all $\gamma \in X$ . Since $|\varphi (\alpha )|=F(\alpha )<h(\alpha )$ , there is some $\gamma \in X$ such that $\gamma \notin \varphi (\alpha )$ , and thus $u=t^\frown \gamma $ is as desired.

Fact 1.12. If $h=^*h'$ , then ${\mathbb S^{h}_\kappa }$ and ${\mathbb S^{h'}_\kappa }$ are forcing equivalent.

Proof Since ${\mathbb S^{h}_\kappa }\cap {\mathbb S^{h'}_\kappa }$ is dense in both ${\mathbb S^{h}_\kappa }$ and ${\mathbb S^{h'}_\kappa }$ .

Definition 2.1. Let A be a set of ordinals and $\mathbb P_\xi $ a forcing notion for each $\xi \in A$ . Let $\mathcal C$ be the set of functions p with $\mathrm {dom}(p)=A$ such that $p(\xi )\in \mathbb P_\xi $ for each $\xi \in A$ . For any $p\in \mathcal C$ , the support of p is defined as

We define the ${<}\lambda $ -support product as follows:

$$ \begin{align*} \overline{\mathbb P}=\textstyle \prod_{\xi\in A}\mathbb P_\xi={\left\{p\in\mathcal C\ \middle |\ \left|\mathrm{supp}(p)\right|< \lambda\right\}}. \end{align*} $$

This is a forcing poset under the ordering $q\leq _{\overline {\mathbb P}} p$ iff $q\leq _{\mathbb P_\xi }p$ for all $\xi \in A$ . Generally we will say “ ${\leq }\kappa $ -support” instead of “ ${<}\kappa ^+$ -support”.

Proof If $\left \langle p_\alpha \mid \alpha \in \gamma \right \rangle $ is a descending sequence in $\overline {\mathbb S}$ and $\gamma \in \kappa $ , then $\textstyle \bigwedge \!_{\alpha \in \gamma }p_\alpha $ is a condition below each $p_\alpha $ , since each ${\mathbb S^{h_\xi }_\kappa }$ is ${<}\kappa $ -closed.

Definition 2.3. Given $p,q\in \overline {\mathbb S}$ , $\alpha \in \kappa $ , and $Z\subseteq A$ with $|Z|<\kappa $ , let $q\leq _{Z,\alpha }p$ iff $q\leq p$ and for each $\xi \in Z$ we have $q(\xi )\leq _\alpha p(\xi )$ .

A generalised fusion sequence is a sequence $\left \langle (p_\alpha ,Z_\alpha )\mid \alpha \in \kappa \right \rangle $ such that:

1. (1) $p_\alpha \in \overline {\mathbb S}$ and $Z_\alpha \in [A]^{<\kappa }$ for each $\alpha \in \kappa $ .

2. (2) $p_{\beta }\leq _{Z_\alpha ,\alpha }p_\alpha $ and $Z_\alpha \subseteq Z_{\beta }$ for all $\alpha \leq \beta \in \kappa $ .

3. (3) For limit $\delta $ we have $Z_\delta =\textstyle \bigcup _{\alpha \in \delta } Z_\alpha $ .

4. (4) $\textstyle \bigcup _{\alpha \in \kappa } Z_\alpha =\textstyle \bigcup _{\alpha \in \kappa }\mathrm {supp}(p_\alpha )$ .

Proof Suppose that $\left \langle (p_\alpha ,Z_\alpha )\mid \alpha \in \kappa \right \rangle $ is a generalised fusion sequence, and let $p=\textstyle \bigwedge \!_{\alpha \in \kappa }p_\alpha $ . Point (4) of Definition 2.3 implies that every $\xi \in \mathrm {supp}(p)$ is an element of $Z_{\eta _\xi }$ for some $\eta _\xi \in \kappa $ . This means that if $\beta \geq \alpha \geq \eta _\xi $ , then $p_\beta (\xi )\leq _\alpha p_\alpha (\xi )$ , and thus $\left \langle p_\alpha (\xi )\mid \alpha>\eta _\xi \right \rangle $ is a fusion sequence in ${\mathbb S^{h_\xi }_\kappa }$ . Since ${\mathbb S^{h_\xi }_\kappa }$ is closed under fusion sequences (Lemma 1.7), we can conclude that

$$ \begin{align*} p(\xi)= \textstyle\bigcap_{\alpha\in\kappa} p_\alpha(\xi)\in{\mathbb S^{h_\xi}_\kappa}. \end{align*} $$

Since $\mathrm {supp}(p)=\textstyle \bigcup _{\alpha \in \kappa } Z_\alpha $ , we see that $|\mathrm {supp}(p)|\leq \kappa $ , thus we can conclude that $p\in \overline {\mathbb S}$ .

Proof Note that Fact 1.12 implies that we can assume without loss of generality that $h_\xi \leq F$ for each $\xi \in A$ . Let $p\in \overline {\mathbb S}$ and $\dot f$ be a name such that $p\Vdash _{\overline {\mathbb S}}\text {"}\,\dot f:\check \kappa \to \check \kappa \,\text {"}$ , then we will construct a name $\dot \varphi $ and a condition $p'\leq p$ such that $p'\Vdash \text {"}\,\dot \varphi \in (\mathrm {Loc}_{\check F})^{\mathbf V[\dot G\restriction \check B]}\,\text {"}$ .

The proof is essentially the same as the proof of Theorem 1.8, except that we work with generalised fusion sequences and have to construct a name $\dot \varphi $ for the appropriate F-slalom in $\mathbf V[G\restriction B]$ , since such a slalom is not generally present in the ground model. That is, we will construct a sequence $\left \langle (p_\xi ,Z_\xi )\mid \xi \in \kappa \right \rangle $ with each $p_\xi \in \overline {\mathbb S}$ that is a generalised fusion sequence in $\overline {\mathbb S}$ and names $\dot D_\xi $ for sets of ordinals $D_\xi \in \mathbf V[G\restriction B]$ with $|D_\xi |\leq F(\xi )$ , such that $p_{\xi +1}\Vdash \text {"}\,\dot f(\check \xi )\in \dot D_\xi \,\text {"}$ .

For each $\xi \in \kappa $ and $\beta \in Z_\xi $ we will make sure that $p_\xi (\beta )\in ({\mathbb S^{h_\beta }_\kappa })^*$ is sharp. To start, we let $p_0= p$ and we let $Z_0=\varnothing $ . At limit stages $\delta $ we can define $p_\delta '=\textstyle \bigwedge \!_{\xi \in \delta }p_\xi $ and let $p_\delta \leq p_\delta '$ be defined elementwise such that $p_\delta (\beta )=(p_\delta '(\beta ))^*$ is sharp for each $\beta \in Z_\delta $ .

Suppose we have defined $p_\xi \in \overline {\mathbb S}$ and $Z_\xi $ and that $|Z_\xi |\leq |\xi |$ . As in the proof of Theorem 1.8, we will consider the successor nodes of the $\xi $ -th splitting level, find subtrees that decide the value of $\dot f(\check \xi )$ , and glue the subtrees together. However, in this situation we have to deal with multiple trees at once, namely with each $p_\xi (\beta )$ such that $\beta \in Z_\xi $ . For each $\beta \in Z_\xi $ we define the set of successor nodes of the $\xi $ -th splitting level of $p_\xi (\beta )$ :

$$ \begin{align*} V^\beta_\xi= \textstyle\bigcup{\left\{\mathrm{suc}(u,p_\xi(\beta))\ \middle |\ u\in \mathrm{Split}_\xi(p_\xi(\beta))\right\}.} \end{align*} $$

To deal with $p_\xi (\beta )$ for all $\beta \in Z_\xi $ simultaneously, we have to consider combinations of elements of $V^\beta _\xi $ for $\beta \in Z_\xi $ , and for each combination we will define a condition that decides $\dot f(\check \xi )$ . These combinations are given by functions $g:Z_\xi \to \textstyle \bigcup _{\beta \in Z_\xi }V^\beta _\xi $ with the property that $g(\beta )\in V^\beta _\xi $ . We will refer to such g as choice functions, since g chooses an element of $V^\beta _\xi $ for each $\beta \in Z_\xi $ .

Let $\mathcal V_\xi $ be the set of choice functions on $\{V_\xi ^\beta \mid \beta \in Z_\xi \}$ and $\mathcal V_\xi '$ the set of choice functions on $\{V_\xi ^\beta \mid \beta \in Z_\xi \setminus B\}$ , that is, $\mathcal V_\xi '$ is the set of $g\restriction (Z_\xi \setminus B)$ with $g\in \mathcal V_\xi $ .

By induction hypothesis $p_\xi (\beta )\in ({\mathbb S^{h_\beta }_\kappa })^*$ for each $\beta \in Z_\xi \setminus B$ , hence we know that

$$ \begin{align*} |\mathrm{Split}_\xi(p_\xi(\beta))|\leq h_\beta(\xi)^{|\xi|}\leq F(\xi), \end{align*} $$

and thus $|V_\xi ^\beta |\leq F(\xi )$ for all $\beta \in Z_\xi \setminus B$ . Since we assume that $|Z_\xi |\leq |\xi |$ , we therefore have $|\mathcal V_\xi '|\leq F(\xi )^{|\xi |}=F(\xi )$ (assuming without loss of generality that $F(\xi )$ is infinite). Hence, if we restrict our attention to $Z_\xi \setminus B$ , we have a small number of choice functions. Consequently, we can describe a name $\dot D_\xi $ depending only on the support in B, i.e., $\dot D_\xi $ names a set in $\mathbf V[G\restriction B]$ , such that $\dot D_\xi $ is bounded in cardinality by $F(\xi )$ .

For any choice function $g\in \mathcal V_\xi $ , let $(p_\xi )_g$ be the condition defined by

$$ \begin{align*} (p_\xi)_g(\beta)=\begin{cases}p_\xi(\beta), &\text{if }\beta\notin Z_\xi,\\ ({p_\xi(\beta)})_{g(\beta)}, &\text{if }\beta\in Z_\xi. \end{cases} \end{align*} $$

Here $({p_\xi (\beta )})_{g(\beta )}$ is the subtree of $p_\xi (\beta )$ generated by the initial segment $g(\beta )\in V^\beta _\xi $ .

Let $\zeta =|\mathcal V_\xi |$ then $\zeta <\kappa $ by inaccessibility of $\kappa $ . Fix some enumeration $\left \langle g_\eta \mid \eta \in \zeta \right \rangle $ of $\mathcal V_\xi $ , which we will use to recursively define a decreasing sequence of conditions $r_\eta $ with $r_\eta \leq _{Z_\xi ,\xi }p_\xi $ for each $\eta \in \zeta $ . Essentially, our recursive construction will result in $r_{\eta +1}$ being like $r_\eta $ , except that $(r_\eta )_{g_\eta }$ is replaced by a stronger condition that decides $\dot f(\check \xi )$ . At the end of the recursion, we will be left with a condition $r_\zeta $ such that $(r_\zeta )_{g}$ decides $\dot f(\check \xi )$ for every $g\in \mathcal V_\xi $ . We then gather the possible values of $\dot f(\check \xi )$ to construct the name $\dot D_\xi $ .

Let $r_0=p_\xi $ . For limit $\delta \in \zeta $ let $r_\delta =\textstyle \bigwedge \!_{\eta \in \delta }r_\eta $ , which is a condition by ${<}\kappa $ -closure (Lemma 2.2). Assuming that $r_\eta \leq _{Z_\xi ,\xi }p_\xi $ for each $\eta \in \delta $ , it is easy to see that $r_\delta \leq _{Z_\xi ,\xi }p_\xi $ as well.

Suppose $r_\eta $ is defined and $r_\eta \leq _{Z_\xi ,\xi } p_\xi $ , then in particular $r_\eta (\beta )\leq _\xi p_\xi (\beta )$ for all $\beta \in Z_\xi $ , and thus $\mathrm {Split}_\xi (r_\eta (\beta ))=\mathrm {Split}_\xi (p_\xi (\beta ))$ for all $\beta \in Z_\xi $ . Therefore by definition of the ordering on ${\mathbb S^{h_\beta }_\kappa }$ and the fact that $p_\xi (\beta )$ is sharp, we see that $V_\xi ^\beta $ is exactly the set of successors of nodes at the $\xi $ -th splitting level of $r_\eta (\beta )$ . Take the $\eta $ -th choice function $g_\eta \in \mathcal V_\xi $ , and let $r^{\prime }_{\eta }\leq (r_\eta )_{g_\eta }$ be such that $r^{\prime }_{\eta }\Vdash \text {"}\,\dot f(\check \xi )=\check \beta ^\eta _\xi \,\text {"}$ for some ordinal $\beta ^\eta _\xi $ . We define $r_{\eta +1}$ elementwise.

If $\beta \notin Z_\xi $ , then we simply take $r_{\eta +1}(\beta )=r_{\eta }'(\beta )$ .

If $\beta \in Z_\xi $ , fix some $w\in \mathrm {suc}(u,r^{\prime }_{\eta }(\beta ))$ for some $u\in \mathrm {Split}_\xi (r^{\prime }_{\eta }(\beta ))$ and consider the subtree $(r^{\prime }_{\eta }(\beta ))_w$ generated by the initial segment w. Now we are ready to define $r_{\eta +1}(\beta )$ as

$$ \begin{align*} r_{\eta+1}(\beta)=(r^{\prime}_{\eta}(\beta))_w\ \cup\ {\left\{u\in r_{\eta}(\beta)\ \middle |\ \exists v\in V_\xi^\beta\setminus \{g_\eta(\beta)\}\ (u\subseteq v\text{ or }v\subseteq u)\right\}}. \end{align*} $$

In words, $r_{\eta +1}(\beta )$ is the result of replacing the extensions of $g_\eta (\beta )\in r_\eta (\beta )$ by $(r^{\prime }_\eta (\beta ))_w$ that decides $\dot f(\check \xi )$ , where we use the subtree $(r^{\prime }_\eta (\beta ))_w$ instead of $r_\eta '(\beta )$ to make sure that $r_{\eta +1}(\beta )$ has enough successors at each splitting level to be in ${\mathbb S^{h_\beta }_\kappa }$ (compare this to the role of $(T^v)_w$ instead of $T^v$ in the proof of Theorem 1.8).

To finish the construction of the next condition in the fusion sequence, we use ${<}\kappa $ -closure to define $p_{\xi +1}'=\textstyle \bigwedge \!_{\eta \in \zeta }\ r_\eta $ and let $p_{\xi +1}=(p^{\prime }_{\xi +1})^*$ be sharp. To see that $p_{\xi +1}\leq _{Z_\xi ,\xi }p_\xi $ , note that for every $\beta \in Z_\xi $ and $v\in V^\beta _\xi $ we have $v\in r_\eta (\beta )$ for all $\eta \in \zeta $ , hence $v\in p_{\xi +1}(\beta )$ . This implies by definition of $V^\beta _\xi $ that $p_{\xi +1}(\beta )\leq _\xi p_\xi (\beta )$ for all $\beta \in Z_\xi $ . Finally, we can let $Z_{\xi +1}=Z_\xi \cup {\left \{{\delta }\right \}}$ for some ordinal $\delta $ , using bookkeeping to make sure that $\textstyle \bigcup _{\xi \in \kappa }Z_\xi =\textstyle \bigcup _{\xi \in \kappa }\mathrm {supp}(p_\xi )$ .

Note that the set of conditions $r\leq p_{\xi +1}$ with $|r(\beta )\cap V_\xi ^\beta |=1$ for all $\beta \in Z_\xi $ , is dense below $p_{\xi +1}$ . For any such r, let g map $\beta $ to the unique element of $r(\beta )\cap V_\xi ^\beta $ for each $\beta \in Z_\xi $ , then $g\in \mathcal V_\xi $ is a choice function, so we see that there exists $\eta \in \zeta $ such that $g=g_\eta $ . We will show that $r\leq r_\eta '$ , which implies that $r\Vdash \text {"}\,\dot f(\check \xi )=\check \beta _\xi ^\eta \,\text {"}$ .

For any $\beta $ we have $r(\beta )\leq p_{\xi +1}(\beta )\leq r_{\eta +1}(\beta )$ . If $\beta \notin Z_\xi $ , then we simply have $r_{\eta +1}(\beta )=r_\eta '(\beta )$ , thus we are done. Otherwise $\beta \in Z_\xi $ , and we know that $g(\beta )$ is an initial segment of the stem of $r(\beta )$ , hence

$$ \begin{align*} r(\beta)=(r(\beta))_{g(\beta)}\subseteq (r_{\eta+1}(\beta))_{g(\beta)}= (r^{\prime}_\eta(\beta))_w, \end{align*} $$

where w is as in the definition of $r_{\eta +1}(\beta )$ above. Since $r(\beta )\leq r_{\eta +1}(\beta )$ , we also have

$$ \begin{align*} r(\beta)=(r(\beta))_w\leq(r_{\eta+1}(\beta))_w=(r^{\prime}_\eta(\beta))_w\leq r^{\prime}_\eta(\beta), \end{align*} $$

and thus $r(\beta )\leq r^{\prime }_\eta (\beta )$ .

We are now ready to construct the names $\dot D_\xi $ such that

$$ \begin{align*} p_{\xi+1}\Vdash\text{"}\,\dot f(\check\xi)\in \dot D_\xi\text{ and }\dot D_\xi\in \mathbf V[\dot G\restriction\check B]\text{ and }|\dot D_\xi|\leq \check F(\check\xi)\,\text{"}. \end{align*} $$

For any $g\in \mathcal V_\xi $ , we define

$$ \begin{align*} g"&=g\restriction (Z_\xi\cap B),\\ E_g&={\left\{{\eta\in\zeta\mid \exists g'\in \mathcal V_\xi'(g'\cup g"=g_\eta)}\right\}},\\ D_\xi^g&={\left\{{\beta_\xi^\eta\mid \eta\in E_g}\right\}}. \end{align*} $$

Since $|\mathcal V_\xi '|\leq F(\xi )$ , we see that $|E_g|\leq F(\xi )$ , hence $|D_\xi ^g|\leq F(\xi )$ . Clearly, if $g,\tilde g\in \mathcal V_\xi $ and $g\restriction (Z_\xi \cap B)=\tilde g\restriction (Z_\xi \cap B)$ , then $D_\xi ^g=D_\xi ^{\tilde g}$ .

Let $\mathcal A_\xi $ be an antichain below $p_{\xi +1}$ such that $r\in \mathcal A_\xi $ implies $|r(\beta )\cap V_\xi ^\beta |=1$ for all $\beta \in Z_\xi $ , and let $g_r\in \mathcal V_\xi $ be such that $g_r(\beta )$ is the single element of $r(\beta )\cap V_\xi ^\beta $ for each $\beta \in Z_\xi $ . We define

$$ \begin{align*} \dot D_\xi={\left\{{(r,\check D_\xi^{g_r})\mid r\in \mathcal A_\xi}\right\}}. \end{align*} $$

It is clear by the above that for each $r\in \mathcal A_\xi $ and $\eta $ such that $g_r=g_\eta $ we have

$$ \begin{align*} r\Vdash \text{"}\,\dot f(\check\xi)=\check \beta_\xi^\eta\in \check D_\xi^{g_r}\text{ and }|\check D_\xi^{g_r}|\leq\check F(\check \xi)\,\text{"}, \end{align*} $$

so by denseness

$$ \begin{align*} p_{\xi+1}\Vdash\text{"}\,\dot f(\check\xi)\in \dot D_\xi\text{ and }|\dot D_\xi|\leq \check F(\check \xi)\,\text{"}. \end{align*} $$

To see that $p_{\xi +1}\Vdash \text {"}\, \dot D_\xi \in \mathbf V[\dot G\restriction \check B]\,\text {"}$ , we argue within $\mathbf V[ G\restriction B]$ . For every $r,\tilde r\in \mathcal A_\xi $ such that both $r\restriction B$ and $\tilde r\restriction B$ are elements of $G\restriction B$ we see that the corresponding $g_r$ and $g_{\tilde r}$ have the property that $g_r\restriction (Z_\xi \cap B)=g_{\tilde r}\restriction (Z_\xi \cap B)$ , and therefore $D_\xi ^{g_r}=D_\xi ^{g_{\tilde r}}$ . Thus, we can fix any arbitrary such $r\in \mathcal A_\xi $ for which $r\restriction B\in G\restriction B$ holds, and see that

$$ \begin{align*} \mathbf V[G\restriction B]\vDash \text{"}\,p_{\xi+1}\restriction B^c\Vdash\dot D_\xi =\check D_\xi^{g_r}\,\text{"}. \end{align*} $$

Let $p'=\textstyle \bigwedge \!_{\xi \in \kappa }p_\xi $ be the limit of the generalised fusion sequence, and let $\dot \varphi $ be a name such that $p'\Vdash \text {"}\,\dot \varphi :\check \xi \mapsto \dot D_\xi \,\text {"}$ , then $\dot \varphi $ names an F-slalom in $\mathbf V[G\restriction B]$ and $p'\Vdash \text {"}\,\dot f\in ^*\dot \varphi \,\text {"}$ .

Proof The lemma is trivial if $|B|\leq \kappa ^+$ , so we will assume that $|B|\geq \kappa ^{++}$ .

We work in $\mathbf V[G]$ . Let $\mu <|B|$ and let ${\left \{{\varphi _\xi \mid \xi \in \mu }\right \}}$ be a family of F-slaloms, then we want to describe some $f\in {}^{\kappa }\kappa $ such that $f\notin ^*\varphi _\xi $ for each $\xi \in \mu $ . Since $\overline {\mathbb S}$ is ${<}\kappa ^{++}$ -c.c., we could find $A_\xi \subseteq A$ with $|A_\xi |\leq \kappa ^+$ for each $\xi \in \mu $ such that $\varphi _\xi \in \mathbf V[G\restriction A_\xi ]$ . Since $|B|>\mu \cdot \kappa ^+$ , we may fix some $\beta \in B\setminus \textstyle \bigcup _{\xi \in \mu }A_\xi $ for the remainder of this proof. Let $f=\textstyle \bigcap _{p\in G}p(\beta )$ , then $f\in {}^{\kappa }\kappa $ is the generic $\kappa $ -real added by the $\beta $ -th term of the product $\overline {\mathbb S}$ .

Continuing the proof in the ground model, let $\dot f$ be an $\overline {\mathbb S}$ -name for f and $\dot \varphi _\xi $ be an $\overline {\mathbb S}$ -name for $\varphi _\xi $ , let $p\in \overline {\mathbb S}$ and $\alpha _0\in \kappa $ . We want to find some $\alpha \geq \alpha _0$ and $q\leq p$ such that $q\Vdash \text {"}\,\dot f(\check \alpha )\notin \dot \varphi _\xi (\check \alpha )\,\text {"}$ .

Let $C={\left \{{\alpha \in \kappa \mid p(\beta )\cap {}^\alpha \kappa =\mathrm {Split}_\alpha (p(\beta ))}\right \}}$ , which is a club set by Lemma 1.10. Since $S_\beta $ is stationary, there exists some $\alpha \geq \alpha _0$ such that $\alpha \in C\cap S_\beta $ . Choose some $p_0\leq p$ such that $p_0(\beta )=p(\beta )$ and such that there is a $Y\in [\kappa ]^{\leq F(\alpha )}$ for which $p_0\Vdash \text {"}\,\dot \varphi _\xi (\check \alpha )=\check Y\,\text {"}$ . This is possible, since $\varphi _\xi \in \mathbf V[G\restriction A_\xi ]$ and $\beta \notin A_\xi $ , therefore we could find $p_0'\in \overline {\mathbb S}\restriction A_\xi $ with $p_0'\leq p\restriction A_\xi $ and Y with the aforementioned property, and then let $p_0(\eta )=p^{\prime }_0(\eta )$ if $\eta \in A_\xi $ and $p_0(\eta )=p(\eta )$ otherwise.

Each $t\in p_0(\beta )\cap {}^\alpha \kappa $ is a $h_\beta (\alpha )$ -splitting node, hence the set $X=\left \{\chi \in \kappa \mid t^\frown \chi \in p_0(\beta )\right \}$ has cardinality $|X|\geq h_\beta (\alpha )$ . Because $\alpha \in S_\beta $ and $\beta \in B$ , we have by our assumptions on F that $|Y|\leq F(\alpha )<h_\beta (\alpha )\leq |X|$ . We can therefore find some $\chi \in X$ such that $\chi \notin Y$ . Let $q\leq p_0$ be defined as

$$ \begin{align*} q(\eta)=\begin{cases} (p_0(\beta))_{t^\frown \chi},&\text{if }\eta=\beta,\\ p_0(\eta),&\text{otherwise}. \end{cases} \end{align*} $$

Here $(p_0(\beta ))_{t^\frown \chi }$ is the subtree of $p_0(\beta )$ generated by the initial segment $t^\frown \chi $ . Then $q\leq p_0\leq p$ and $q\Vdash \text {"}\,\dot f(\check \alpha )\notin \check Y=\dot \varphi _\xi (\check \alpha )\,\text {"}$ .

Proof This is a standard argument of counting names.

Proof We assume that $\mathbf V\vDash \text {"}\,2^\kappa =\kappa ^+\,\text {"}$ , or otherwise we first use a forcing to collapse $2^\kappa $ to become $\kappa ^+$ . By a result of Solovay (Theorem 8.10 in [Reference Jech5]) there exists a family of $\kappa $ many disjoint stationary subsets of $\kappa $ , thus let ${\left \{{S_\eta \mid \eta \in \kappa }\right \}}$ be such a family. Let $\kappa \leq \gamma \in \kappa ^+$ and $\sigma :\kappa \to \gamma $ be given. We will assume without loss of generality that $\sigma $ is bijective, and hence that $\sigma ^{-1}:\gamma \to \kappa $ is a well-defined bijection. Let $\left \langle \lambda _\xi \mid \xi \in \gamma \right \rangle $ be an increasing sequence of cardinals with $\mathrm {cf}(\lambda _\xi )>\kappa $ for all $\xi \in \gamma $ .

Fix some $F\in {}^{\kappa }\kappa $ such that $F(\alpha )^{|\alpha |}=F(\alpha )$ and $2^{F(\alpha )}\leq F(\beta )$ for any $\alpha <\beta $ . For each $\eta \in \kappa $ we define a function $g_\eta $ as follows:

$$ \begin{align*} g_\eta(\alpha)=\begin{cases} F(\alpha),&\text{ if }\alpha\in S_\eta,\\ 2^{F(\alpha)},&\text{ otherwise}. \end{cases} \end{align*} $$

For each $\xi \in \gamma $ we define $H_\xi \in {}^{\kappa }\kappa $ as follows:

$$ \begin{align*} H_\xi(\alpha)=\begin{cases} F(\alpha),&\text{ if }\alpha\in\textstyle\bigcup_{\zeta\in\xi}S_{\sigma^{-1}(\zeta)},\\ 2^{F(\alpha)},&\text{ otherwise}. \end{cases} \end{align*} $$

For each $\xi \in \gamma $ let $A_\xi $ be a set of ordinals with $|A_\xi |=\lambda _\xi $ , such that $\left \langle A_\xi \mid \xi \in \gamma \right \rangle $ is a sequence of mutually disjoint sets, and let $A=\textstyle \bigcup _{\xi \in \gamma } A_\xi $ . For each $\xi \in \gamma $ and $\beta \in A_\xi $ , we define $h_\beta =H_{\xi }$ .

We now consider the product forcing $\overline {\mathbb S}=\prod _{\beta \in A}{\mathbb S^{h_\beta }_\kappa }$ with ${\leq }\kappa $ -support. Let G be $\overline {\mathbb S}$ -generic. We will fix some $\eta \in \kappa $ , and let $B=\textstyle \bigcup _{\xi \in {\sigma (\eta )}+1}A_{\xi }$ and $B^c=A\setminus B$ . By Lemma 2.8 we see that $(\mathrm {Loc}_{g_\eta })^{\mathbf V[G\restriction B]}$ has cardinality

$$ \begin{align*}\kappa^+\cdot |B|=\kappa^+\cdot \left|\textstyle\sup_{\xi\leq {\sigma(\eta)}}A_{\xi}\right|=\kappa^{+}\cdot\left|A_{\sigma(\eta)}\right|=\lambda_{\sigma(\eta)}. \end{align*} $$

To use Lemma 2.5, we need that $h_\beta \leq ^* g_\eta $ for all $\beta \in B^c$ , equivalently, that $H_{\xi }\leq ^*g_\eta $ for all $\xi \in ({\sigma (\eta )},\gamma )$ . But this is true for any $\xi \in ({\sigma (\eta )},\gamma )$ , since $g_\eta =F(\alpha )$ iff $\alpha \in S_\eta =S_{\sigma ^{-1}({\sigma (\eta )})}$ and because ${\sigma (\eta )}\in \xi $ we see that $H_{\xi }(\alpha )=F(\alpha )$ as well. Meanwhile for all $\alpha \notin S_\eta $ we have $g_\eta (\alpha )=2^{F(\alpha )}\geq H_{\xi }(\alpha )$ . Therefore Lemma 2.5 shows that $(\mathrm {Loc}_{g_\eta })^{\mathbf V[G\restriction B]}$ is a family in $\mathbf V[G]$ of size $\lambda _{\sigma (\eta )}$ that forms a witness for

$$ \begin{align*} \mathbf V[G]\vDash\text{"}\,{\mathfrak d_\kappa^{g_\eta}(\in^*)}\leq \lambda_{\sigma(\eta)}\,\text{"}. \end{align*} $$

On the other hand, if $\beta \in A_{{\sigma (\eta )}}$ , then $h_\beta =H_{{\sigma (\eta )}}$ and thus for any $\alpha \in S_\eta =S_{\sigma ^{-1}({\sigma (\eta )})}$ we see that $g_\eta (\alpha )<H_{\sigma (\eta )}(\alpha )$ . Therefore by Lemma 2.7 we see that

$$ \begin{align*} \mathbf V[G]\vDash\text{"}\,\lambda_{\sigma(\eta)}=|A_{\sigma(\eta)}|\leq {\mathfrak d_\kappa^{g_\eta}(\in^*)}\,\text{"}. \end{align*} $$

In conclusion, we get for every $\eta \in \kappa $ that

$$ \begin{align*} \mathbf V[G]\vDash\text{"}\,\lambda_{\sigma(\eta)}={\mathfrak d_\kappa^{g_\eta}(\in^*)}\,\text{"}.\\[-32pt] \end{align*} $$

Definition 3.1. Given a sequence $\mathcal S={\left \{{S_\eta \mid \eta \in \kappa ^+}\right \}}$ of almost disjoint subsets of $\kappa $ and a partition ${\left \{{X,Y}\right \}}$ of $\kappa ^+$ , we define ${\mathbb W^{X,Y}_\kappa }$ to have tuples $p=(s_p,A_p,B_p)$ as conditions, where $s_p\in [\kappa ]^{<\kappa }$ and $A_p\in [X]^{<\kappa }$ and $B_p\in [Y]^{<\kappa }$ are such that

$$ \begin{align*} \textstyle\bigcup_{\eta\in A_p}S_\eta\cap \textstyle\bigcup_{\eta\in B_p}S_\eta\subseteq \sigma_{s_p}. \end{align*} $$

The ordering on ${\mathbb W^{X,Y}_\kappa }$ is given by $(s_q,A_q,B_q)\leq (s_p,A_p,B_p)$ if all of the following hold:

1. (i) $A_p\subseteq A_q$ ,

2. (ii) $B_p\subseteq B_q$ ,

3. (iii) $s_p=s_q\cap \sigma _{s_p},$

4. (iv) $s_q\cap [\sigma _{s_p},\sigma _{s_q})\supseteq \textstyle \bigcup _{\eta \in A_p}S_\eta \cap [\sigma _{s_p},\sigma _{s_q})$ ,

5. (v) $s_q\cap \textstyle \bigcup _{\eta \in B_p}S_\eta \subseteq \sigma _{s_p}$ .

Proof Let $p\in {\mathbb W^{X,Y}_\kappa }$ . It is clear from the way we have defined the forcing that for any $\eta \in A_p$ we have $p\Vdash \text {"}\,\check S_\eta \subseteq ^*\dot z_G\,\text {"}$ and for any $\eta \in B_p$ we have $p\Vdash \text {"}\,\check S_\eta \cap \dot z_G\subseteq \check \sigma _p\in \kappa \,\text {"}$ . Therefore, we are done if we prove that:

1. (1) for every $\eta \in X$ there is $q\leq p$ such that $\eta \in A_q$ , and

2. (2) for every $\eta \in Y$ there is $q\leq p$ such that $\eta \in B_q$ .

Proving (1) and (2) happens in the same way, so we only prove (1) below.

Fix some $\eta \in X$ . Since $S_\eta $ is almost disjoint from $S_\xi $ for all $\xi \in B_p$ , we can define $\gamma _\xi \in \kappa $ such that $S_\xi \cap S_\eta \subseteq \gamma _\xi $ for each $\xi \in B_p$ . Since $|B_p|<\kappa $ we see that $\textstyle \bigcup _{\xi \in B_p}\gamma _\xi \in \kappa $ . Pick some $\gamma \in \textstyle \bigcup _{\xi \in A_p}S_\xi $ such that $\gamma \geq \sigma _{s_p}\cup \textstyle \bigcup _{\xi \in B_p}\gamma _\xi $ .

We define $q\leq p$ by

$$ \begin{align*} s_q&=s_p\cup \left( \textstyle\bigcup_{\xi\in A_p}S_\xi\cap [\sigma_{s_p},\gamma]\right),\\ A_q&=A_p\cup{\left\{{\eta}\right\}},\\ B_q&=B_p. \end{align*} $$

Note that $\gamma \in s_q$ , thus $\sigma _{s_q}=\gamma +1$ . Furthermore, note that

$$ \begin{align*} \textstyle\bigcup_{\xi\in A_p}S_\xi\cap \textstyle\bigcup_{\xi\in B_p}S_\xi\subseteq \sigma_{s_p}&\leq\gamma,\quad\text{ and}\\ S_\eta\cap \textstyle\bigcup_{\xi\in B_p}S_\xi\subseteq \textstyle\bigcup_{\xi\in B_p}\gamma_\xi&\leq \gamma. \end{align*} $$

Therefore, q is indeed a condition.

Proof Let $\gamma \in \kappa $ be limit and let $\left \langle p_\eta \mid \eta <\gamma \right \rangle $ be a descending sequence of conditions. We will write $p_\eta =(s_\eta ,A_\eta ,B_\eta )$ . Let $p=(s_p,A_p,B_p)$ be given by $s_p=\textstyle \bigcup _{\eta \in \gamma }s_\eta $ and $A_p=\textstyle \bigcup _{\eta \in \gamma }A_\eta $ and $B_p=\textstyle \bigcup _{\eta \in \gamma }B_\eta $ . That p is a condition and that $p\leq p_\eta $ for each $\eta \in \gamma $ are easy to check.

Proof See, for example, Lemma 23.7 in [Reference Jech5].

Definition 3.5. Let $\mathbb P$ be a forcing. We will call a set $P\subseteq \mathbb P$ centred if for every $Q\in [P]^{<\kappa }$ there exists $q\in \mathbb P$ such that $q\leq p$ for all $p\in Q$ . We say that $\mathbb P$ is $\kappa $ -centred if $\mathbb P=\textstyle \bigcup _{\alpha \in \kappa }P_\alpha $ where each $P_\alpha $ is centred.

Proof For any $s\in [\kappa ]^{<\kappa }$ and $A\in [X]^{<\kappa }$ we define

$$ \begin{align*} W_{s,A}={\left\{{p\in{\mathbb W^{X,Y}_\kappa}\mid s_p=s\land A_p=A}\right\}}. \end{align*} $$

Since $|X|\leq \kappa $ implies that $\left |[\kappa ]^{<\kappa }\times [X]^{<\kappa }\right |=\kappa $ , we are done if we show that each $W_{s,A}$ is centred. Let $Q\in [ W_{s,A}]^{<\kappa }$ and $B=\textstyle \bigcup _{p\in Q}B_p$ . We claim that $q=\left \langle s,A,B\right \rangle $ is a condition and that $q\leq p$ for all $p\in Q$ .

Suppose that $\alpha \in \textstyle \bigcup _{\eta \in A} S_\eta \cap \textstyle \bigcup _{\eta \in B}S_\eta $ , then there is $p\in Q$ such that $\alpha \in \textstyle \bigcup _{\eta \in A} S_\eta \cap \textstyle \bigcup _{\eta \in B_p}S_\eta $ , and since p is a condition it follows that $\alpha \in \sigma _{s_p}=\sigma _s$ . Hence q is a condition. To check that $q\leq p$ for each $p\in Q$ , note that (i)–(iii) of Definition 3.1 are immediate, while (iv) and (v) hold vacuously by $s_p=s_q$ .

Proof Note that each term ${\mathbb W^{X_\xi ,Y_\xi }_\kappa }$ is ${<}\kappa $ -closed and ${<}\kappa ^+$ -c.c. (the latter as a consequence of $\kappa $ -centredness), thus a ${<}\kappa $ -support product of length $\kappa ^+$ is also ${<}\kappa $ -closed, which is easily proved, and ${<}\kappa ^+$ -c.c., which is proved using a $\Delta $ -system argument. That $2^\kappa =\kappa ^+$ will remain true, follows from an argument by counting names, using that $|{\mathbb W^{X_\xi ,Y_\xi }_\kappa }|=\kappa ^+$ for each $\xi \in \kappa ^+$ , and that the product has $\kappa ^+$ many terms, thus $|\overline {\mathbb W}|=\kappa ^+$ .

Proof We start with a model $\mathbf V\vDash \text {"}\,2^\kappa =\kappa ^+\text { and }\Diamond _\kappa \,\text {"}$ containing a sad family $\mathcal S=\left \langle S_\eta \mid \eta \in \kappa ^+\right \rangle $ , and we will assume without loss of generality that $\sigma :\kappa ^+\to \kappa ^+$ is a bijection. We define the functions $g_\eta $ for each $\eta \in \kappa ^+$ as

$$ \begin{align*} g_\eta(\alpha)=\begin{cases} F(\alpha), & \text{if }\alpha\in S_\eta,\\ 2^{F(\alpha)}, & \text{otherwise}. \end{cases} \end{align*} $$

For each $\eta \in \kappa ^+$ we define the partition ${\left \{{X_\eta ,Y_\eta }\right \}}$ of $\kappa ^+$ by

$$ \begin{align*} X_\eta&=\sigma^{-1}[\sigma(\eta)]={\left\{{\zeta\in\kappa^+\mid \sigma(\zeta)\in\sigma(\eta)}\right\}},\\ Y_\eta&=\kappa^+\setminus X_\eta. \end{align*} $$

We then force with a ${<}\kappa $ -support product $\overline {\mathbb W}=\prod _{\eta \in \kappa ^+}{\mathbb W^{X_\eta ,Y_\eta }_\kappa }$ . Note, in particular, that $|X_\eta |\leq \kappa $ . Let G be $\overline {\mathbb W}$ -generic, then we will work in $\mathbf V[G]$ . We define $z_ {\sigma (\eta )}=\textstyle \bigcup _{p\in G}s_{p(\eta )}$ , then $z_{\sigma (\eta )}$ is ${\mathbb W^{X_\eta ,Y_\eta }_\kappa }$ -generic over $\mathbf V$ .

By Lemma 3.8, we know that $\mathbf V[G]\vDash \text {"}\,2^\kappa =\kappa ^+\text { + }\check {\mathcal S}\text { is a sad family}\,\text {"}$ . Therefore by Lemma 3.2, if $\eta \in \kappa ^+$ , then we have $S_\zeta \subseteq ^* z_{\sigma (\eta )}$ for all $\zeta \in X_\eta $ and $|S_\zeta \cap z_{\sigma (\eta )}|<\kappa $ for all $\zeta \in Y_\eta $ . Equivalently, using the definition of $X_\eta $ and $Y_\eta $ , if $\xi \in \kappa ^+$ , then we have $S_\zeta \subseteq ^* z_\xi $ for all $\zeta \in \kappa ^+$ such that $\sigma (\zeta )\in \xi $ and $|S_\zeta \cap z_{\xi }|<\kappa $ for all $\zeta \in \kappa ^+$ such that $\sigma (\zeta )\in [\xi ,\kappa ^+)$ .

For each $\xi \in \kappa ^+$ we define $H_\xi \in {}^{\kappa }\kappa $ as follows:

$$ \begin{align*} H_\xi(\alpha)=\begin{cases} F(\alpha),&\text{ if }\alpha\in z_\xi,\\ 2^{F(\alpha)},&\text{ otherwise}. \end{cases} \end{align*} $$

The remainder of the proof mirrors the proof of Theorem 2.9 almost exactly.

For each $\xi \in \kappa ^+$ let $A_\xi $ be a set of ordinals with $|A_\xi |=\lambda _\xi $ , such that $\left \langle A_\xi \mid \xi \in \kappa ^+\right \rangle $ is a sequence of mutually disjoint sets, and let $A=\textstyle \bigcup _{\xi \in \kappa ^+} A_\xi $ . For each $\xi \in \kappa ^+$ and $\beta \in A_\xi $ , we define $h_\beta =H_{\xi }$ .

We now consider the product forcing $\overline {\mathbb S}=\prod _{\beta \in A}{\mathbb S^{h_\beta }_\kappa }$ with ${\leq }\kappa $ -support. Let K be $\overline {\mathbb S}$ -generic. We will fix some $\eta \in \kappa ^+$ , and let $B=\textstyle \bigcup _{\xi \in \sigma (\eta )+1}A_{\xi }$ and $B^c=A\setminus B$ . By Lemma 2.8 we see that $(\mathrm {Loc}_{g_\eta })^{\mathbf V[G][K\restriction B]}$ has cardinality

$$ \begin{align*} \kappa^+\cdot |B|=\kappa^+\cdot \left|\textstyle\sup_{\xi\leq {\sigma(\eta)}}A_{\xi}\right|=\kappa^+\cdot \left|A_{\sigma(\eta)}\right|=\lambda_{\sigma(\eta)}. \end{align*} $$

To use Lemma 2.5, we need that $h_\beta \leq ^* g_\eta $ for all $\beta \in B^c$ , equivalently, that $H_{\xi }\leq ^*g_\eta $ for all $\xi \in ({\sigma (\eta )},\kappa ^+)$ . But this is true for any $\xi \in ({\sigma (\eta )},\kappa ^+)$ , since $g_\eta =F(\alpha )$ iff $\alpha \in S_\eta $ , while $H_{\xi }(\alpha )=F(\alpha )$ iff $\alpha \in z_{\xi }$ , and because ${\sigma (\eta )}\in \xi $ we have $S_\eta \subseteq ^*z_{\xi }$ . Therefore Lemma 2.5 shows that $(\mathrm {Loc}_{g_\eta })^{\mathbf V[G][K\restriction B]}$ is a family in $\mathbf V[G][K]$ of size $\lambda _{\sigma (\eta )}$ that forms a witness for

$$ \begin{align*} \mathbf V[G][K]\vDash\text{"}\,{\mathfrak d_\kappa^{g_\eta}(\in^*)}\leq \lambda_{\sigma(\eta)}\,\text{"}. \end{align*} $$

On the other hand, if $\beta \in A_{{\sigma (\eta )}}$ , then $h_\beta =H_{{\sigma (\eta )}}$ and thus $\sigma (\eta )\in [{\sigma (\eta )},\kappa ^+)$ implies that $|S_\eta \cap z_{\sigma (\eta )}|<\kappa $ . In particular, $S_\eta \setminus z_{\sigma (\eta )}$ is stationary and if $\alpha \in S_\eta \setminus z_{\sigma (\eta )}$ , then $g_\eta (\alpha )<H_{\sigma (\eta )}(\alpha )$ . Hence by Lemma 2.7 we see that

$$ \begin{align*} \mathbf V[G][K]\vDash\text{"}\,\lambda_{\sigma(\eta)}=|A_{\sigma(\eta)}|\leq {\mathfrak d_\kappa^{g_\eta}(\in^*)}\,\text{"}. \end{align*} $$

In conclusion, we get for every $\eta \in \kappa $ that

$$ \begin{align*} \mathbf V[G][K]\vDash\text{"}\,\lambda_{\sigma(\eta)}={\mathfrak d_\kappa^{g_\eta}(\in^*)}\,\text{"}.\\[-32pt] \end{align*} $$

Question 4.1. Is it consistent that there exists a family of functions ${\left \{{h_\xi \mid \xi \in \kappa ^{++}}\right \}}$ such that each ${\mathfrak d_\kappa ^{h_\xi }(\in ^*)}$ has a distinct value? Is it consistent that there is a model with $2^\kappa $ many distinct values for cardinals of the form ${\mathfrak d_\kappa ^{h}(\in ^*)}$ ?

Question 4.2. Is it consistent that there exist $h_0,h_1,h_2\in {}^{\kappa }\kappa $ such that $|h_0(\alpha )|<|h_1(\alpha )|<2^{|h_0(\alpha )|}=h_2(\alpha )$ and ${\mathfrak d_\kappa ^{h_2}(\in ^*)}<{\mathfrak d_\kappa ^{h_1}(\in ^*)}<{\mathfrak d_\kappa ^{h_0}(\in ^*)}$ ?

Question 4.3. Do there exist functions $h,h'$ such that $\mathfrak b_\kappa ^{h}(\in ^*)<\mathfrak b_\kappa ^{h'}(\in ^*)$ is consistent?