Proof For each $\beta <\lambda ^+$ , $\dot {\mathbb {Q}}_\beta $ is forced to have size $(\lambda ^{<\lambda })^{V[\mathbb {M}\ast \mathbb {P}_\beta ]}=\lambda $ . Hence $\dot {\mathbb {Q}}_\beta $ is trivially forced to be $\lambda ^+$ -cc. Additionally, the iteration $\mathbb {P}_{\lambda ^+}$ takes direct limits on the stationary set $\lambda ^+\cap \mathrm {cof}(\lambda )$ . By standard arguments (such as in [Reference Baumgartner and Mathias2]), it follows that $\mathbb {P}_{\lambda ^+}$ is $\lambda ^+$ -c.c.

Proof For all $\beta <\alpha $ we claim that $\Vdash _{\mathbb {P}_\beta } \text {“}\mathrm{CU}(\dot T_\beta )$ is $\kappa ^+$ -closed”: if $\langle c_\xi \,|\, \xi <\tau \rangle \subset \mathrm{CU}(T_\beta )$ is strictly decreasing and $\tau <\kappa ^+$ , then $\sup \{\max c_\xi \,|\, \xi <\tau \}$ has cofinality $\mathrm {cf}(\tau )\le \tau <\kappa ^+$ , and so $\bigcup _{\xi <\tau }c_\xi \cup \{\sup \{\max c_\xi \,|\, \xi <\tau \} \}$ is trivially a condition in $\mathrm{CU}(T_\beta )$ . Hence the iteration $\mathbb {P}_\alpha $ is $\kappa ^+$ -closed since $\kappa ^+ < \lambda $ and the supports of the conditions are sufficiently large.

Proof We argue for any $\alpha <\lambda ^+$ and prove (1) and (2) for $\alpha $ at the same time. Specifically, we prove (1) by constructing a lift in two steps, and part of the proof will show that $M[G][H_\alpha ]^{<\lambda } \subset M[G]$ , so (2) will follow because $H_\alpha $ is arbitrary.

First, we lift the embedding to $M[G]$ . Now work in $V[G]$ . Since $\mathbb {P}_\alpha $ is $\kappa ^+$ -closed by Lemma 4.2 and has size $\lambda $ , the Absorption Lemma for Mitchell Forcing implies that $j(\mathbb {M})/G \times \mathbb {P}_\alpha \simeq j(\mathbb {M})/G$ . Let $G'$ be a $j(\mathbb {M})/G$ -generic over $V[G][H_\alpha ]$ ; then there is a $j(\mathbb {M})/G$ -generic $G''$ over $V[G]$ such that $V[G][H_\alpha ][G']=V[G][G'']$ . Let us denote $\tilde G=G\times G''$ .

Since j is the identity below $\lambda $ and conditions in $\mathbb {M}(\kappa ,\lambda )$ are bounded below $\lambda $ , $j[G]=G \subset G\times G''=\tilde G$ , and so we can apply Fact 2.17 to lift $j:M[G] \to N[\tilde G]$ . Now we perform the second step of the lift to extend j to have domain $M[G][H_\alpha ]$ .

Proof of Claim. We argue, working in $N[\tilde G]$ , that there is $q \in j(\mathbb {P}_\alpha )$ that is a lower bound for $j[H_\alpha ]$ : Let $\langle C_\beta \,|\, \beta <\alpha \rangle $ be the sequence of clubs added by $H_\alpha $ , where $C_\beta $ is added at the $(\beta +1)^{\text {st}}$ step of the iteration. Observe that the $C_\beta $ ’s are closed and unbounded in $\lambda $ , the latter because $\lambda \cap \mathrm {cof}(\le \!\kappa )$ is unbounded in $\lambda $ . Let $\mathrm {dom}(q) = j[\alpha ]$ and for $\beta <\alpha $ , let $q(j(\beta )) = C_\beta \cup \{\lambda \}$ , noting that $q\in N[\tilde G]$ since j and $H_\alpha $ are in $N[\tilde G]$ . It will be immediate that q is a lower bound of $j[H_\alpha ]$ once we verify that for $\beta <\alpha $ , $q\!\restriction \! j(\beta )\Vdash ^{N[\tilde {G}]}_{j(\mathbb {P}_\beta )} \text {“}C_\beta \cup \{\lambda \}\text { is a condition in }j(\mathrm {CU}(\dot T_\beta ))$ .”

First note that for all $\beta <\lambda $ , $q\!\restriction \! j(\beta ) \Vdash ^{N[\tilde {G}]}_{j(\mathbb {P}_\beta )}$ “ $C_\beta $ is a closed unbounded set in $j(\dot {T}_\beta )\cap \lambda $ ” since $q\!\restriction \! j(\beta ) \Vdash ^{N[\tilde {G}]}_{j(\mathbb {P}_\beta )}$ “ $j(\dot {T}_\beta )\cap \lambda =T_\beta $ ”; therefore to verify that q is a condition in $j(\mathbb {P}_\alpha )$ it suffices to show that $q\!\restriction \! j(\beta )\Vdash ^{N[\tilde {G}]}_{j(\mathbb {P}_\beta )} \text {“}j(\dot S_\beta ) \cap \lambda \text { is stationary} \text{in }\lambda $ .”

Since $q\!\restriction \! j(\beta )\Vdash ^{N[\tilde {G}]}_{j(\mathbb {P}_\beta )} $ “ $j(\dot S_\beta ) \cap \lambda = S_\beta $ , it remains to argue that $N[\tilde G] \models \text {“}S_\beta $ is stationary after forcing with $j(\mathbb {P}_\beta )$ .” First note that $S_\beta $ is still stationary in $N[G][H_\alpha ]$ by Fact 2.10 because $N[G][H_\alpha ]$ is a generic extension of $N[G][H_\beta ]$ by a $\kappa ^+$ -closed forcing.Footnote ⁶ Second, $S_\beta $ is stationary after forcing with $(j(\mathbb {M})/G)^{N[G]}$ (by Lemma 3.4), so it is stationary in $N[G][H_\alpha ][G']=N[\tilde G]$ . Furthermore, by elementarity of the embedding $j:M[G] \to N[\tilde G]$ , we have $N[\tilde G] \models \text {“}j(\mathbb {P}_\beta )$ is $\kappa ^+$ -closed,” so $ N[\tilde G] \models \text {“}j(\mathbb {P}_\beta )$ preserves the stationarity of $S_\beta $ ” by Fact 2.10. ⊣

Let $\tilde H_\alpha $ be any $j(\mathbb {P}_\alpha )$ -generic over $N[\tilde G]$ that contains the master condition q from Claim 4.4. This tells us that $j[H_\alpha ] \subset \tilde H_\alpha $ , that we can extend to $j:M[G][ H_\alpha ] \to N[\tilde G][ \tilde H_\alpha ]$ by Fact 2.17, and that $M[G]$ and $M[G][H_\alpha ]$ have the same $<\lambda $ -sequences of ordinals by Fact 2.19. Hence we have (2) for $\alpha $ , as explained above.

To finish the proof it suffices to show that cofinal branches are not added to $\lambda $ -trees in the extension of $N[\tilde G \ast \tilde H_\alpha ]$ over $N[G \ast H_\alpha ]$ . If T is a $\lambda $ -tree in $N[G][H_\alpha ]$ , then there are no new cofinal branches in $N[G][H_\alpha ][G']=N[\tilde G]$ by Lemma 3.4. Note that in $N[\tilde G]$ , $\lambda $ is an ordinal of cofinality $\kappa ^+$ , and therefore the tree T has height of cofinality $\kappa ^+$ and levels of size $\kappa ^+$ . However, Fact 2.11 still applies since $2^\kappa>\kappa ^+$ in $N[\tilde G]$ and $j(\mathbb {P}_\alpha )$ is $\kappa ^+$ -closed in $N[\tilde G]$ by elementarity, and we are finished.

Proof By Fact 2.23 it is enough to show that $V[\mathbb {M}] \models \text {“}\mathbb {P}_\alpha $ is $\lambda $ -distributive” for all $\alpha < \lambda ^+$ . Moreover, by choosing a supposed counterexample $\dot f$ , it is enough to prove that $M[\mathbb {M}] \models \text {“}\mathbb {P}_\alpha $ is $\lambda $ -distributive” for a $\lambda $ -sized transitive M with $\dot f$ and $\mathbb {P}_\alpha $ in M, which follows from Lemma 4.3(2).

Proof If $\dot S$ is a $\mathbb {P}_{\lambda ^+}$ -name for a stationary subset of $\lambda \cap \mathrm {cof}(\le \! \kappa )$ , then Fact 2.23 implies that without loss of generality, there is some $\alpha ' < \lambda ^+$ such that $\dot S$ is a $\mathbb {P}_{\alpha '}$ -name. By construction, it follows that for some $\beta ,\gamma <\lambda ^+$ , $\dot S$ is the $\gamma ^{\mathrm{th}}\,\mathbb {P}_{\beta }$ -name for such a stationary set, so $\dot S = \dot S_\alpha $ for some $\alpha < \lambda ^+$ . As observed in the proof of Claim 4.4, the generic subset of $\lambda $ added by the next step of the iteration is club in $\lambda $ , so $V[\mathbb {M} \ast \mathbb {P}_{\alpha +1}] \models $ “There is a club $C \subset \lambda $ such that every $\rho \in C \cap \mathrm {cof}(\kappa ^+)$ is a reflection point of S.” Then $V[\mathbb {M} \ast \mathbb {P}_{\lambda ^+}]$ still satisfies this statement by distributivity of $\mathbb {P}_{\lambda ^+}$ over $V[\mathbb {M}]$ , and hence distributivity of the quotient over $V[\mathbb {M} \ast \mathbb {P}_{\alpha +1}]$ .

Proof By the $\lambda ^+$ -chain condition, the name for any supposed $\lambda $ -Aronszajn tree T in $V[\mathbb {M} \ast \mathbb {P}_{\lambda ^+}]$ would be contained in $V[\mathbb {M}\ast \mathbb {P}_\alpha ]$ for some $\alpha <\lambda ^+$ , and so T would be contained in $M[\mathbb {M} \ast \mathbb {P}_\alpha ]$ for some $\lambda $ -sized transitive model M. Let $j:M \to N$ witnesses weak compactness. By Lemma 4.3, if we fix generics G and $H_\alpha $ , we can lift the embedding to $j:M[G][ H_\alpha ] \to N[\tilde G][ \tilde H_\alpha ]$ . So $N[\tilde G][ \tilde H_\alpha ] \models \text {“}j(T)$ is a $j(\lambda )$ -tree” by elementarity, and any element $t \in j(T)_\lambda $ defines a cofinal branch $\{s \in T: s \le _{j(T)} t \}$ in T. Because of the preservation properties in Lemma 4.3(1), this branch is already contained in $N[G][ H_\alpha ]$ , so it is contained in $V[G][ H_\alpha ]$ .

Proof Instead of going through the whole argument, we will explain here how the proof of Lemma 4.3 can be modified for our present purposes. The crux is as follows: Starting from $V[G][H_\alpha ]$ , we must find a $j(\mathbb {M})$ -generic $\tilde G$ such that $V[G][H_\alpha ][I][G']=V[\tilde {G}]$ where I is $\mathrm {Add}(\kappa ,\lambda ^*)$ -generic over $V[G][H_\alpha ]$ ( $\lambda ^*$ being the least inaccessible in N above $\lambda $ ) and $G'$ is $j(\mathbb {M})/(G *(H_\alpha \times I))$ -generic over $V[G][H_\alpha ][I]$ . However, instead of using the Absorption Lemma, we use the guessing properties of $\ell $ .

To see how we can obtain such a filter $G'$ , first observe that $\dot {\mathbb {P}}_\alpha =j(\ell )(\lambda )$ is a $j(\mathbb {M})\!\restriction \!\lambda =\mathbb {M}$ -name for a $\kappa ^+$ -closed poset. Thus, $j(\mathbb {M})$ is isomorphic to a dense subset of

$$ \begin{align*}j(\mathbb{M})\!\restriction\!\lambda^*\ast \dot{\mathbb{N}}_{\lambda^*}\cong \mathbb{M}\ast(\dot{\mathbb{P}}_\alpha\times\mathrm{Add}(\kappa,\lambda^*))\ast \dot{\mathbb{N}}_{\lambda^*}. \end{align*} $$

Hence we can choose any I that is $\mathrm {Add}(\kappa ,\lambda ^*)$ -generic over $V[G][H_\alpha ]$ and any $G'$ that is $\mathbb {N}_{\lambda ^*}$ -generic over $V[G][H_\alpha ][I]$ , and we let $\tilde G := G \ast (H_\alpha \times I) \ast G'$ .

As in Claim 4.4, the main point is to inductively show that for $\beta <\alpha $ , the $S_\beta $ ’s remain stationary after forcing with the master condition over $N[\tilde G]$ . The argument here is almost identical, except that we now require an application of Fact 2.9 to show that $S_\beta $ remains stationary in the extension from $N[G][H_\alpha ]$ to $N[G][H_\alpha \times I]$ . Then the $G'$ , since it is again a projection of a product of a $\kappa ^+$ -cc and $\kappa ^+$ -closed poset, acts analogously to the $G'$ from the proof of Lemma 4.3. The last step using the $\kappa ^+$ -closure of $j(\mathbb {P}_\alpha )$ is also the same.

Moving forward, we choose $\tilde H_\alpha $ to be a generic containing the master condition to define the extension $N[\tilde G \ast \tilde H_\alpha ] = N[G][H_\alpha ][I][G'][\tilde H_\alpha ]$ . The last way in which this argument differs from that for Lemma 4.3 is in showing that the extension from $N[G \ast H_\alpha ]$ to $N[\tilde G \ast \tilde H_\alpha ]$ does not add cofinal branches to $\lambda $ -trees. In our present case we must again pay attention to the extension $N[G \ast H_\alpha ][I]$ over $N[G \ast H_\alpha ]$ , so we use an extra application of Fact 2.12.

Proof Suppose for contradiction that ${\sf AP}(\kappa ^{++})$ holds in some extension by $\mathbb {M}\ast \dot {\mathbb {P}}$ . We may fix an enumeration $\vec {a}=\langle a_i:i<\lambda \rangle \in V[\mathbb {M} \ast \dot {\mathbb {P}}_{\lambda ^+}]$ of bounded subsets of $\lambda $ and find a club $C\subseteq \lambda $ in $V[\mathbb {M} \ast \dot {\mathbb {P}}_{\lambda ^+}]$ so that every $\gamma \in C$ is approachable with regard to $\vec {a}$ . By Fact 2.23, there is some $\alpha <\lambda ^+$ such that $\vec {a}, C \in V[\mathbb {M} \ast \dot {\mathbb {P}}_\alpha ]$ .

Next, we will make use of Lemma 4.8. Let M be a transitive $\lambda $ -sized model containing $\mathbb {P}_\alpha , \ell , \vec {a}$ , and C. Since $\ell $ witnesses the weakly compact Laver diamond for $\lambda $ , we find N such that $j:M \to N$ witnesses weak compactness of $\lambda $ and $j(\ell )(\lambda ) = \dot {\mathbb {P}}_\alpha $ . Use Lemma 4.8 to obtain an embedding $j:M[G\ast H_\alpha ]\to N[\tilde G\ast \tilde H_\alpha ]$ for some N, where $G\ast H_\alpha $ is $\mathbb {M}\ast \dot {\mathbb {P}}_\alpha $ -generic over V. Note that $j(C)\cap \lambda =C$ and that $j(\vec {a})\!\restriction \!\lambda =\vec {a}$ . Consequently, $\lambda $ is a limit point of $j(C)$ and hence a member of $j(C)$ , and by definition of $j(C)$ , we have that $\lambda $ is approachable with respect to $\vec {a}$ in $N[\tilde G\ast \tilde H_\alpha ]$ . Since $N[\tilde G \ast \tilde {H}_\alpha ] \models $ “ $\mathrm {cf}(\lambda )=\kappa ^+$ ,” we find a club $e\subseteq \lambda $ of order-type $\kappa ^+$ witnessing this. By definition, we then have that every initial segment of e is in the range of $\vec {a}$ , and hence every initial segment of e is a member of $M[G\ast H_\alpha ]$ .

Let U denote the tree $(\lambda ^{<\kappa ^+})^{V[G\ast H_\alpha ]}$ , and observe that U has width $\lambda ^\kappa =\lambda $ . By the remarks in the previous paragraph, we see that e gives a cofinal branch through U. Moreover, $e\in N[\tilde G \ast {\tilde H}_\alpha ]$ . Note that $V[G \ast H_\alpha ] \models $ “ $\lambda $ is regular,” and this remains true in $V[G \ast H_\alpha \ast I]$ where I is the $\mathrm {Add}(\kappa ,\lambda ^*)$ -generic from Lemma 4.8. Therefore, if we show that $e \in V[G \ast H_\alpha \ast I]$ , we obtain a contradiction since the order-type of e is $\kappa ^+$ .

Now we argue that $e \in V[G \ast H_\alpha \ast I]$ . By the $j(\lambda )$ -distributivity of $j(\mathbb {P}_\alpha )$ , we conclude that $e \in N[\tilde G]$ . However, with the same notation as in Lemma 4.8, we see that in the model $N[G \ast H_\alpha \ast I] \subset N[G \ast H_\alpha \ast I \ast H'] = N[\tilde G]$ , we have that $2^\kappa>\lambda $ , meaning that $2^\kappa $ is greater than the width of U. Hence adjoining $H'$ does not add a branch through U because in $N[G \ast H_\alpha \ast I]$ , $\mathbb {N}_{\lambda ^*}$ is a projection of a product of a $\kappa ^+$ -closed forcing and a square- $\kappa ^+$ -cc forcing. Therefore e is a member of $V[G\ast H_\alpha \ast I]$ , a contradiction.

Fact 4.15. Suppose that $\gamma <\lambda $ is a limit, that t is a normal, homogeneous $\gamma $ -tree with all levels of size $<\lambda $ , and that $b\subseteq t$ is a cofinal branch. Then there exists a condition $t'\in \mathbb {S}$ of height $\gamma +1$ extending t so that $b \in t'$ .

Fact 4.16. Let $\dot {\mathbb {T}}$ be the $\mathbb {S}$ -name for the forcing $(\dot {T},<_{\dot {T}})$ . Then $:$

1. (1) $\mathbb {S}$ is $\lambda $ -strategically closed and hence $\lambda $ -distributive.

2. (2) $\mathbb {S}\ast \dot {\mathbb {T}}$ has a dense, $\lambda $ -closed subset.

3. (3) $\mathbb {S}\ast \dot {\mathbb {T}}$ is forcing-equivalent to $\mathrm {Add}(\lambda ,1)$ .

Proof We know that in $V[\mathbb {M}]$ , $\mathbb {S}\ast \dot {\mathbb {T}}$ is $\kappa ^+$ -closed (in fact, $\lambda $ -closed) on a dense subset. Furthermore, $\mathbb {T}$ is $\kappa ^+$ -distributive after forcing with $\mathbb {S}$ , so $\mathbb {P}_\alpha $ , which is $\kappa ^+$ -closed after forcing with $\mathbb {S}$ , remains $\kappa ^+$ -closed after forcing with $\mathbb {S}\ast \dot {\mathbb {T}}$ . Thus $\mathbb {K}_\alpha \simeq \mathbb {S}\ast \dot {\mathbb {T}}\ast \dot {\mathbb {P}}_\alpha $ is $\kappa ^+$ -closed on a dense subset.

Proof Assuming (3) for $\beta <\alpha $ , we prove (1) and (2) for $\alpha $ at the same time. Specifically, part of the proof will show that $M[G][T][H_\alpha ]^{<\lambda } \subset M[G][T]$ , so (2) follows because $H_\alpha $ is arbitrary. Then we will use the type of lift constructed in (1) for $\alpha $ to prove (3) for $\alpha $ .

Proof of (1) and (2). We will lift the embedding $j:M \to N$ in three steps.

Our first step is to lift $j:M \to N$ to have domain $M[G]$ . Since $j(\ell )(\lambda )=\dot {\mathbb {K}}_\alpha $ is a $j(\mathbb {M})\!\restriction \!\kappa =\mathbb {M}$ -name for a $\kappa ^+$ -closed poset, there is a dense subset of $j(\mathbb {M})$ giving us the forcing-equivalence

$$ \begin{align*}j(\mathbb{M})\simeq \mathbb{M}\ast(\dot{\mathbb{K}}_\alpha \times\mathrm{Add}(\kappa,\lambda^*))\ast\dot{\mathbb{N}}_{\lambda^*}, \end{align*} $$

where $\lambda ^*$ is the least N-inaccessible above $\lambda $ . Fix a $(\mathbb {T}\times \mathbb {P}_\alpha )$ -generic filter $B\times H_\alpha $ over $V[G\ast T]$ . Here B denotes the generic branch through T. Let I be $\mathrm {Add}(\kappa ,\lambda ^*)$ -generic over $V[G \ast T \ast (B \times H_\alpha )]$ , so that $T \ast (B \times H_\alpha \times I)$ is $\mathbb {K}_\alpha \ast \mathrm {Add}(\kappa ,\lambda ^*)$ -generic over $V[G]$ . Let $G'$ be $\mathbb {N}_{\lambda ^*}$ -generic over $V[G \ast T \ast ((B \times H_\alpha ) \times I)]$ . Since $j[G]=G\subseteq \tilde G$ , we may lift j to an elementary embedding $j:M[G]\to N[\tilde G]$ working in $N[G \ast T \ast ((B \times H_\alpha ) \times I) \ast G']=N[j(G)]=N[\tilde G]$ .

The next step is to lift the embedding j to have domain $M[G \ast T]$ . We claim that there is a condition $t^*\in j(\mathbb {S})$ so that $T\subseteq t^*$ . Indeed, T is a normal, homogeneous $\lambda $ -tree in $N[\tilde G]$ all of whose levels have size $<j(\lambda )$ . Since B is a cofinal branch through T in $N[\tilde G]$ , we may find, by Fact 4.15, a condition in $j(\mathbb {S})$ which extends T. Set $t^*\in j(\mathbb {S})$ to be the minimal extension of T by B. Let $\tilde T$ be $j(\mathbb {S})$ -generic over $V[\tilde G]$ so that $t^*\in \tilde T$ . Since $T=j[T]$ and T is an initial segment of $t^*$ , we may extend j to an elementary embedding $j:M[G\ast T]\to N[\tilde G\ast \tilde T]$ .

The third step in the construction of the embedding is to show that we can lift the embedding j to have domain $M[G \ast T \ast H_\alpha ]$ .

Proof of Claim. This is analogous to the proof of Claims 4.4 and 4.9; instead of going through all the details, we will point to the differences. Once again, the main point is to prove that if $\beta <\alpha $ , then $S_\beta $ remains stationary in $N[\tilde G\ast \tilde T]$ . By the inductive assumption that (3) holds for $\beta <\alpha $ , T is still Suslin after forcing with $\mathbb {P}_\beta $ , and hence $\mathbb {T}$ is still $\lambda $ -cc in $V[G\ast T\ast H_\beta ]$ . Thus $S_\beta $ is stationary in $V[G\ast T\ast (B\times H_\beta )]$ (here $H_\beta :=H_\alpha \cap \mathbb {P}_\beta $ ). The tail of the forcing $\mathbb {P}_\alpha $ from stage $\beta $ onwards is still $\kappa ^+$ -closed after forcing with $\mathbb {T}$ , since $\mathbb {T}$ is $\lambda $ -distributive. Since $\kappa ^{<\kappa }=\kappa $ in $V[G\ast T\ast H_\beta ]$ and $S_\beta $ consists of points of cofinality $\leq \kappa $ , $S_\beta $ is stationary in $V[G\ast T \ast B \ast H_\alpha \ast I]$ and hence in $N[G\ast T \ast B \ast H_\alpha \ast I]$ too. The argument that $S_\beta $ remains stationary in $N[\tilde G]$ (after adjoining $G'$ ) is analogous to the previous ones. Finally, we use the $j(\lambda )$ -distributivity of $j(\mathbb {S})$ to show that $S_\beta $ remains stationary in $N[\tilde G \ast \tilde T]$ . ⊣

As in the proof of Lemma 4.3, we choose $\tilde H_\alpha $ to contain the master condition to complete the lift and to obtain $M[G \ast T \ast H_\alpha ]^{<\lambda } \subset M[G \ast T]$ and hence (2). ⊣

Proof of (3). Suppose $A \subset T$ is a maximal antichain with $A \in V[G \ast T \ast H_\alpha ]$ , and find a model M such that $A \in M[G \ast T \ast H_\alpha ]$ . Using the lift $j:M[G \ast T \ast H_\alpha ] \to N[\tilde G \ast \tilde T \ast \tilde H_\alpha ]$ , we will show that $j(A)$ is bounded in $\tilde T$ . It will then follow by elementarity that A is bounded in T.

Now let $\dot A$ be a $\mathbb {P}_\alpha $ -name for A in $V[G][ T]$ such that $\dot A \in M[G][T]$ . Observe that by the elementarity of j, $j(A)\!\restriction \!\lambda =A$ ; in particular, A is a member of $N[\tilde G\ast \tilde T\ast \tilde H_\alpha ]$ . Recall that $t^*$ is the minimal extension of T by the branch B. We will show that every node of $t^*$ at level $\lambda $ extends some element of A. This shows that A is in fact a maximal antichain in $\tilde T$ , and hence $j(A)=A$ , since $j(A)\supseteq A$ is also a maximal antichain in $\tilde T$ .

We recall that $t^*:=T \cup \{ s^{\frown } (B\setminus \gamma ):s\in T\wedge \gamma <\lambda \}.$ By definition of $t^*$ , to show that every element of $t^*$ on level $\lambda $ extends some element of A, it suffices to show that the following holds:

• (*) for each $s\in T$ and $\gamma <\lambda $ , there is some $\eta>\gamma $ so that $s^{\frown } (B\!\restriction \![\gamma ,\eta ))$ extends an element of A.

By upwards absoluteness, it suffices to verify that $(*)$ holds in $V[G\ast T\ast (B\times H_\alpha )]$ ; this will use a density argument in $V[G \ast T]$ and the genericity of the branch B. We will now work over the model $V[G\ast T]$ to argue that for each $s\in T$ and $\gamma <\lambda $ , the set

$$ \begin{align*}\{ (u,r)\in\mathbb{T}\times\mathbb{P}_\alpha:(\exists\eta>\gamma)\; (u,r)\Vdash s^{\frown}(\dot{B}\!\restriction\![\gamma,\eta))\text{ extends a node in }\dot{A}\}, \end{align*} $$

denoted $D_{s,\gamma }$ , is dense in $\mathbb {T}\times \mathbb {P}_\alpha $ .

So fix some condition $(u,r)$ in $\mathbb {T}\times \mathbb {P}_\alpha $ as well as a node $s\in T$ and an ordinal $\gamma <\lambda $ . Extend u if necessary so that lh $(u)\geq \gamma $ , noting that $u\Vdash _{\mathbb {T}}\dot {B}\!\restriction \!\gamma =u\!\restriction \!\gamma $ . Set $\bar {u}:=u\!\restriction \![\gamma ,\text {lh}(u))$ (it is possible that this is the empty sequence). Since $u\in \mathbb {T}$ , u is an element of T. By the homogeneity of T, $\bar {u}\in T$ . As $s\in T$ , we have by another use of homogeneity that $s^{\frown }\bar {u}\in T$ . Recalling that $\dot {A}$ is forced by $\mathbb {P}_\alpha $ to be a maximal antichain in T, we may extend r to a condition $r^*$ in $\mathbb {P}_\alpha $ and find a (possibly trivial) extension $s^{\frown }\bar {u}^{\frown }\bar {u}^*$ of $s^{\frown }\bar {u}$ in T so that $r^*\Vdash s^{\frown }\bar {u}^{\frown }\bar {u}^*$ extends an element of $\dot {A}$ . Now set $u^*:=u^{\frown } \bar {u}^*=(u\!\restriction \!\gamma )^{\frown }\bar {u}^{\frown }\bar {u}^*$ , and let $\eta $ be the length of $u^*$ . We observe that $u^*\Vdash \dot {B}\!\restriction \![\gamma ,\eta )=\bar {u}^{\frown }\bar {u}^*$ . From this it now follows that $(u^*,r^*)$ forces that $s^{\frown }(\dot {B}\!\restriction \![\gamma ,\eta ))$ extends an element of $\dot {A}$ , which completes the proof that $D_{s,\gamma }$ is dense. This in turn shows that $(*)$ holds in $V[G\ast T\ast (B\times H_\alpha )]$ .

In summary, since $(*)$ holds, we conclude that every node in $t^*$ on level $\lambda $ extends an element of A. As shown earlier, we now see that $j(A)=A$ , and hence $j(A)$ is bounded in $\tilde T$ . By the elementarity of j, A is bounded in T, completing the proof that T is Suslin after forcing with $\mathbb {P}_\alpha $ . ⊣

This completes the proof of Lemma 4.18.

Proof As in Section 4.1, this uses the $\lambda ^+$ -cc of the full iteration $\mathbb {P}$ together with the lifting lemma. The difference is that we use the definition of $\ell $ to specifically use a transitive N and an elementary $j:M\to N$ with critical point $\lambda $ so that $j,M\in N$ and so that $j(\ell )(\lambda )=\dot {\mathbb {K}}_\alpha $ .

Proof This is very similar to the analogous lemma for Theorem 1.2. We find the relevant $\vec a$ and C in $M[G][T][H_\alpha ]$ for some $\alpha <\lambda ^+$ and lift a weakly compact embedding to $j:M[G][T][H_\alpha ] \to N[\tilde G][\tilde T][\tilde H_\alpha ]$ . We can show that for the relevant tree U, we have a cofinal branch $e \in N[\tilde G][\tilde T][\tilde H_\alpha ]$ . Then we use the $j(\lambda )$ -distributivity of $j(\mathbb {S}\ast \dot {\mathbb {P}}_\alpha )$ to show that $e \in N[\tilde G]=N[G][T][B][H_\alpha ][I][G']$ . Since $\mathbb {N}_{\lambda ^*}$ is a projection of a product of a $\kappa ^+$ -closed poset with a square- $\kappa ^+$ -cc poset and $N[G][T][B][H_\alpha ][I] \models $ “ $2^\kappa> \lambda $ ,” we see that $e \in N[G][T][H_\alpha ][B][I]$ . Then because $N[G][T][B][H_\alpha ][I] \models $ “ $\lambda $ is regular,” we have a contradiction.