Proof. First we note that for all $a \in K \setminus H$ , $b,b' \in L \setminus H$ , the word $\varrho (ba, b'a)$ is an alternating word (over the union of $K\setminus H$ and $L \setminus H$ ) of length $6640$ .

1. (A) By Corollary 2.11 (and $R \subseteq R^+$ ), it is enough to argue that $R^+$ satisfies $C'(\frac {1}{10})$ . To this end, fix two elements $g \neq g^*$ in $R^+$ , as well as some canonical representations

   $$ \begin{align*}\begin{array}{lll} g&=&g_0g_1\cdots g_{n-1},\\ g^*&=&g^*_0g^*_1\cdots g_{m-1}^*. \end{array}\end{align*} $$
   By Clause (2) of Observation 2.6, there are $i,i^* \in \omega $ such that $n \in \{6640 n_i, 6640n_i+1\}$ , $m \in \{6640 n_{i^*}, 6640n_{i^*}+1\}$ .

   Let $l \in \omega $ , and assume that

   (*) $$ \begin{align} \bigwedge_{k\le l}(K \ast_H L \models g^{-1}_{k-1}g^{-1}_{k-2}\cdots g_0^{-1}g^*_0g^*_1\cdots g^*_{k-1} \in H), \end{align} $$
   so we have to show that $l < 664 \cdot \min (n_i,n_{i^*})$ .

   Assume on the contrary that $l \geq 664\cdot n_i$ . We can choose $\sigma \in \Sigma $ , ${\varepsilon } \in \{1,-1\}$ , such that letting $r= h_\sigma ^{-1}\varrho _{n_i}(b_\sigma a_\sigma , b_\sigma 'a_\sigma )$ , g is a weakly cyclically reduced conjugate of $r^{\varepsilon } = (h_\sigma ^{-1}\varrho _{n_i}(b_\sigma a_\sigma , b_\sigma 'a_\sigma ))^{\varepsilon }$ if $n_i = 0$ , or of $r^{\varepsilon } = (\varrho _{n_i}(b_\sigma a_\sigma , b_\sigma 'a_\sigma ))^{{\varepsilon }}$ , and similarly for $g^*$ , $r^*$ and $\sigma ^*$ , ${\varepsilon }^*$ . If we fix the canonical representations

   $$ \begin{align*}r = u_0u_1 \cdots u_{6640 n_i-1},\end{align*} $$
   where $u_j \in \{b_\sigma ,b_\sigma ',a_\sigma ,h_\sigma ^{-1}b_\sigma \}$ , and similarly
   $$ \begin{align*}r^* = u^*_0u^*_1 \cdots u^*_{6640 n_{i^*}-1},\end{align*} $$
   then again recalling Observation 2.6(2), we can assume that there exist $j < 6640 n_i$ , $j^* <6640n_{i^*}$ , such that whenever $0 < k < 6640 n_i-1$ , then $g_k = u^{\varepsilon }_{j+ {\varepsilon } k}$ , and if $0<k < 6640 n_{i^*}-1$ , then $g^*_k = (u^*_{j^* + {\varepsilon }^* k})^{{\varepsilon }^*}$ .

   We first observe that $i = i^*$ , since otherwise if, say, $i<i^*$ did hold, then for some $1 \leq k,k' \leq 81 n_i$ with $u_{j + {\varepsilon } k} =b_\sigma $ , $u_{j + {\varepsilon } k'} =b^{\prime }_\sigma $ , while $u^*_{j + {\varepsilon } k} = u^*_{j + {\varepsilon } k'} \in \{b_{\sigma ^*}, b^{\prime }_{\sigma ^*}\}$ , and so by (*), we get

   $$ \begin{align*}b_\sigma = u_{j + {\varepsilon} k} \sim_H u^*_{j + {\varepsilon} k} = u^*_{j + {\varepsilon} k'} \sim_H u_{j + {\varepsilon} k'} =b_\sigma',\end{align*} $$
   contradicting $b_\sigma \not \sim _H b_\sigma '$ . From now on, n will denote the common value of $n_i = n_{i^*}$ .

   Now note that $b_\sigma \sim _H b_{\sigma ^*}$ : there is a k with $1 \leq k \leq 10 n$ such that $u_{j + {\varepsilon } k} \in \{b_\sigma , h^{-1}_\sigma b_\sigma \}$ , and $u^*_{j^* + {\varepsilon }^* k} \in \{b_{\sigma ^*}, h^{-1}_{\sigma ^*}b_{\sigma ^*}\}$ , so by (*) for some $h \in H$ , we have $b^{-{\varepsilon }}_\sigma h b^{{\varepsilon }^*}_{\sigma ^*} \in H$ , implying that $b_\sigma \sim _H b_{\sigma ^*}$ . Similarly, for some $k^\bullet $ , $1 \leq k^\bullet \leq 2 n$ , $u_{j + {\varepsilon } k^\bullet } = a_\sigma $ , and $u^*_{j^* + {\varepsilon }^* k^\bullet } = a_{\sigma ^*}$ , and by the same line of reasoning, $a_\sigma \sim _H a_{\sigma ^*}$ .

   We clearly get that

   $(\boxminus )$ either $(\circleddash )_c$ , or $(\circleddash )_d$ , or $\sigma = \sigma ^*$ holds, and in each case, $b_\sigma \not \sim _H b_{\sigma ^*}'$ .

   Now, note that if $j \neq j^*$ or ${\varepsilon } \neq {\varepsilon }^*$ , then there exists k with $1 \leq k <500 n$ such that $u_{j + {\varepsilon } k} \in \{b_\sigma , h_\sigma ^{-1}b_\sigma \}$ , and $u^*_{j^* + {\varepsilon }^* k} = b^{\prime }_{\sigma ^*} = b_\sigma '$ , and for some $h \in H$ , we have $b_\sigma ^{-{\varepsilon }} h(b^{\prime }_{\sigma ^*})^{{\varepsilon }^*} \in H$ (or $(h_\sigma ^{-1}b_\sigma )^{-{\varepsilon }} h(b^{\prime }_{\sigma ^*})^{{\varepsilon }^*}$ ), so $b_\sigma \sim _H b_{\sigma ^*}'$ , contradicting $(\boxminus )$ . Therefore, hereafter, we can assume that $j = j^*$ and ${\varepsilon } = {\varepsilon }^*$ .

   We now divide our analysis into a few cases and subcases:

   1. ▸ If either $(\circleddash )_c$ or $\sigma = \sigma ^*$ , then necessarily, $b_\sigma = b_{\sigma ^*}$ and $b_\sigma ' = b_{\sigma ^*}'$ . But now for some k with $1\leq k \leq 10n$ , $g_k = g^*_k = b_\sigma ^\varepsilon $ , so for

      $$ \begin{align*}h = g^{-1}_{k-1}g^{-1}_{k-2} \cdots g^{-1}_0g^*_0g^*_1 \cdots g^*_{k-1} \in H,\end{align*} $$
      we have
      $$ \begin{align*}g^{-1}_k h g^*_k \in H,\end{align*} $$
      but then $H \mathrel {\leq _{\mathrm {m}}} L$ together with $b_\sigma \in L \setminus H$ imply that .

      1. ▸▸ If $\sigma = \sigma ^*$ , then invoking Observation 2.6(2) again (and recalling that g and $g^*$ are cyclically reduced conjugates of $h^{-1}_\sigma \varrho (b_\sigma a_\sigma , b_\sigma 'a_\sigma )$ ), it is straightforward to check that $j = j^*$ and ${\varepsilon } = {\varepsilon }^*$ imply $g = g^*$ , which is a contradiction.

      2. ▸▸ If $\sigma \neq \sigma ^*$ and $a_\sigma \neq a_{\sigma ^*}$ , then implies that

         
         and in the following step (conjugating by $b_\sigma =b_{\sigma ^*}$ again), we get a contradiction.

   2. ▸ If the pair $\sigma $ , $\sigma ^*$ satisfies condition $(\circleddash )_d$ , then we argue as follows. First, we claim that the there exists a k with $1 \leq k < 10n+2$ such that following three hold:

      1. (⊠)₁ $g_k = u_{j + {\varepsilon } k}^{\varepsilon } = a^{\varepsilon }_\sigma $ ,

      2. (⊠)₂ $g^*_k = (u^*_{j + {\varepsilon } k})^{\varepsilon } = a^{{\varepsilon }}_{\sigma ^*}$ ,

      3. (⊠)₃ $h = g^{-1}_{k-1}g^{-1}_{k-2} \cdots g^{-1}_0g^*_0g^*_1 \cdots g^*_{k-1} \in H \setminus K' = H \setminus H'$ .

      As before, for some $k^\bullet < 10n$ , we have $u^{\varepsilon }_{j + {\varepsilon } k^\bullet } = a^{\varepsilon }_\sigma $ , and $(u^*_{j + {\varepsilon } k^\bullet })^{\varepsilon } = a^{{\varepsilon }}_{\sigma ^*}$ , $u_{j + {\varepsilon } (k^\bullet +1)}^{\varepsilon } = b^{\varepsilon }_\sigma $ , and $u^*_{j + {\varepsilon } (k^\bullet +1)} = b^{{\varepsilon }}_{\sigma ^*}$ . Suppose that

      $$ \begin{align*}h = g^{-1}_{k^\bullet-1}g^{-1}_{k^\bullet-2} \cdots g^{-1}_0g^*_0g^*_1 \cdots g^*_{k^\bullet-1} \in H'.\end{align*} $$
      Then $h' = a^{-{\varepsilon }}_\sigma h a^{\varepsilon }_{\sigma ^*} \in K' H' K' = K'$ , and by our indirect assumptions $a^{-{\varepsilon }}_\sigma h a^{\varepsilon }_\sigma \in H$ , so $h'$ lies in the intersection $K' \cap H = H'$ . Now
      $$ \begin{align*}u_{j + {\varepsilon} (k^\bullet+1)}^{-{\varepsilon}} h' u^{{\varepsilon}}_{j + {\varepsilon} (k^\bullet+1)} = b^{-{\varepsilon}}_\sigma h' b^{\varepsilon}_{\sigma^*} \in b^{-{\varepsilon}}_\sigma H' b^{{\varepsilon}}_{\sigma^*},\end{align*} $$
      so by $(\circleddash )_d$ (iii), this product is not in $H'$ ; thus, we can assume that some $k<10n +2$ satisfies $(\boxtimes )_1$ – $(\boxtimes )_2$ .But then using $a_\sigma , a_{\sigma ^*} \in K' \setminus H'$ ,
      $$ \begin{align*} g^{-1}_{k}g^{-1}_{k} \cdots g^{-1}_0g^*_0g^*_1 \cdots g^*_{k}&= a_\sigma^{-{\varepsilon}}h a_{\sigma^*}^{{\varepsilon}}\\&\in (K' \setminus H) \cdot (H \setminus H') \cdot (K' \setminus H) \subseteq K \setminus H. \end{align*} $$
      This is a contradiction.

2. (B) Fix , and $z \in M^* \setminus K$ , with a canonical form $z = z_0z_1 \cdots z_{m-1}$ satisfying it does not contain any subsequence $z_{\sigma _0}z_{\sigma _0+1}\cdots z_{\sigma _0+j-1}$ that is a subsequence of a canonical form of an element $r \in R$ , where $j> \frac {6640}{2}+1$ (we can assume this, since otherwise, we could insert the entire sequence of the inverse of this fixed canonical form of r). Now suppose that holds in $M^*$ ; that is,

   $$\begin{align*}M \models zgz^{-1}g' \in N. \end{align*}$$
   W.l.o.g. $z_0,z_{m-1} \in L$ (thus m is odd), since otherwise, we can replace g with , and $g'$ with . Now as $g,g' \in K$ , if $g,g'$ are not in H, then the product $z_0z_1 \cdots z_{m-1}gz_{m-1}^{-1} \cdots z_0^{-1}g'$ is in a weakly cyclically reduced form; otherwise, if $g \in H$ , then $z_{m-1}gz_{m-1}^{-1} \in L \setminus H$ (here, we use $H \mathrel {\leq _{\mathrm {m}}} L$ and that $z_{m-1} \in L \setminus H$ ), and similarly, $g' \in H$ implies $z_0^{-1}g'z_0 \in L \setminus H$ . So by these reductions, we obtain a product in a cyclically reduced form of length $2m+2$ or $2m$ , or $2m-2$ . A cyclic conjugate of this word contains a long ( $> 7/10$ ) subword of some canonical form of an $r \in R$ . By our assumptions on z (not containing more than half of a canonical representation of r), this has to involve either g or $g'$ ; in fact, either the word $z_{j}z_{j+1} \cdots z_{m-1}gz^{-1}_{m-1} z^{-1}_{m-2} \cdots z^{-1}_{j}$ (or $z_{j}z_{j+1} \cdots z_{m-2} (z_{m-1}gz^{-1}_{m-1}) z^{-1}_{m-2} \cdots z^{-1}_{j}$ if $g \in H$ ), or $z^{-1}_{j_*}z^{-1}_{j_*-1} \cdots z^{-1}_{0}g'z_{0} z_{1} \cdots z_{j_*}$ contains a long ( $>2/10$ fraction) subword of a canonical form of some $r \in R$ (in the latter case, if $g' \in H$ , then of course we mean the word $z^{-1}_{j_*}z^{-1}_{j_*-1} \cdots (z^{-1}_{0}g'z_{0}) z_{1} \cdots z_{j_*}$ ). But this is impossible since in any $r = r_0r_1 \cdots r_{n-1} \in R$ ( $n \in \{6640,6641\}$ ) at any fixed $t \in [\frac {6640}{10}, \frac {6640 \cdot 9}{10}]$ , there exists $k < 250$ such that (for some $\sigma \in \Sigma $ ) $r_{t-k} \in Hb_\sigma ^{\pm 1}H$ , $r_{t+k} \in H(b_\sigma ')^{\pm 1}H$ , and so $r_{t-k} \not \sim _H r_{t+k}$ , while $z_k$ , $z_k^{-1}$ are clearly $\sim _H$ -related.

3. (C) Suppose otherwise, for example, for some $k,k' \in K$ either

   
   or
   
   Observe that after performing the cancellations in the free amalgam M and writing $y = y_0y_1\cdots y_{m-1}$ as a reduced (alternating) word, in both cases (regardless of whether $k,k' \in H$ ), there is at most one j for which $y_j \in L \setminus H$ and $y_j \not \sim _H b$ . Now possibly replacing $y_0y_1 \cdots y_{m-1}$ with a weakly cyclically reduced conjugate of it (if the reduced form of $y_0y_1 \cdots y_{m-1}$ is not weakly cyclically reduced), this clause remains true (and the resulting word similarly belongs to N in M). It is not difficult to see that there exists at least one $j'$ such that $y_{j'} \sim _H b$ . Again, $y_0y_1 \cdots y_{m-1}$ (or a cyclical permutation of it) contains a long subword of a canonical form of some $r \in R$ , but any such subword (if longer than $400$ ) contains at least two-two occurrences of $b_\sigma $ and $b_\sigma '$ (for some $\sigma \in \Sigma $ ), and b cannot be $\sim _H$ -equivalent with both $b_\sigma $ and $b_\sigma '$ (since $b_\sigma \not \sim _H b_\sigma '$ ).

4. (D) This is the same as above. Assuming that $M^* \models bab' \in K$ , then for some $a' \in K$ , , so

   $$ \begin{align*}M \models bab'a' \in N.\end{align*} $$
   Now if $a' \in K \setminus H$ , then the word $bab'a'$ is weakly cyclically reduced, so any weakly cyclically reduced conjugate to it is of length either $4$ or $5$ and clearly cannot contain a long subword of any $r \in R$ .

   If $a' \in H$ , then depending on whether $b" = b'a'b \in H$ , we have that either $b^{-1}(bab'a')b = ah \in K \setminus H$ is weakly cyclically reduced (so $M \models bab'a' = b(ah)b^{-1} \notin bNb^{-1} = N$ ) or $b^{-1}bab'a'b = ab'a'b = a b"$ (where $b" \notin H$ ), which is weakly cyclically reduced, and similarly cannot lie in N.

5. (E) Let $a,a' \in K \setminus H$ be such that $a \not \sim _{H'} a'$ , and fix $l,l' \in L'$ . Suppose that ; that is,

   $$ \begin{align*}M \models w= ala'l' \in N.\end{align*} $$
   We can write w as a reduced word. If $l \in H$ , then $l \in H'$ , and since $a \not \sim _{H'} a'$ , we have $ala' \in K \setminus H'$ , so either $w= (ala')l'$ is a product of an element of $K \setminus H'$ and $L \setminus H$ (if $l' \notin H$ ) which has to lie in $K(L\setminus H)$ (which is disjoint to N) or $(ala')l' \in (K \setminus H') \cdot H' = K \setminus H'$ , and we are done.

   So w.l.o.g. $l \notin H$ . (Similarly, implies that w.l.o.g. $l' \notin H$ ). So any weakly cyclically reduced conjugate of $w \in M$ has length at most $5$ and contains at least $2$ entries from $K \setminus H$ . But $w \in N$ implies that some weakly cyclically reduced conjugate contains a long subword of some $r \in R$ , which is clearly impossible.

6. (F) The proof of (E) works here too.

7. (G) Let $g \in M^*$ , $n \in \omega $ , $n>1$ be such that , . Recalling Observation 2.6, we can write g as an alternating product of elements of $K\setminus H$ and $L \setminus H$

   $$ \begin{align*}g = g_0g_1 \cdots g_{2m-1}.\end{align*} $$
   W.l.o.g. there exists no conjugate $ygy^{-1}$ of g, and $g'$ with $g'(ygy^{-1})^{-1} \in N$ such that $g'$ has a shorter canonical representation than $2m$ , since we can replace g with $g'$ and get a torsion element. Therefore, there is no $r \in R$ , $\sigma _0 <2m$ with the sequence $g_{\sigma _0}g_{\sigma _0+1} \cdots g_{2m-1}g_0g_1 \cdots g_{\sigma _0-1}$ containing a subsequence of a canonical representation of r of length $j> \frac {6640}{2}+1$ .Now, since
   $$ \begin{align*}M \models(g_0g_1 \cdots g_{2m-1})^n \in N,\end{align*} $$
   there exists a cyclic conjugate of $(g_0g_1 \cdots g_{2m-1})^n$ and a subsequence $s_0s_1 \cdots s_j$ of it that is also a subsequence of a canonical form of some $s \in R$ with $j \geq \frac {7}{10} \cdot 6640$ . Our assumptions above on $g_0g_1 \cdots g_{2m-1}$ easily imply
   $$ \begin{align*}2m \leq \frac{6640}{2}+1;\end{align*} $$
   thus,
   $$ \begin{align*}2m + \frac{2}{10}\cdot 6640-1 \leq j,\end{align*} $$
   and clearly, $2m+330 \leq j$ . This way we get that $s_\ell \sim _H s_{\ell +2m}$ for each $\ell \leq 330$ , but as s is a cyclically reduced conjugate of $h_\sigma ^{-1}\varrho (b_\sigma a_\sigma ,b_\sigma 'a_\sigma )$ or of its inverse (for some $\sigma \in \Sigma $ ), we get that for some $\ell \in [1, 330] s_{\ell } \in H(b_\sigma )^{\pm 1}H$ , $s_{\ell +2m} \in H(b^{\prime }_\sigma )^{\pm 1}H$ ; thus, $s_\ell \not \sim _H s_{\ell +2m}$ . This is a contradiction.

Notation 4.2. Whenever the map $d:[\kappa ]^2\rightarrow \theta $ is clear from the context, we define for all $\gamma < \kappa $ and $i \leq \theta $ , the following sets:

• • $D^\gamma _{<i}=\{\beta < \gamma \mathrel {|} d(\beta , \gamma ) < i\}$ , and

• • $D^\gamma _{\le i}=\{\beta < \gamma \mathrel {|} d(\beta , \gamma ) \le i\}$ .

Proof. Suppose that $d:[\kappa ]^2\rightarrow \theta $ is subadditive, and let $\gamma ,i$ and $\beta $ be as above.

$\blacktriangleright $ By Definition 4.1(1), for every $\alpha \in D^\beta _{<i}$ , $d(\alpha ,\gamma )\le \max \{ d(\alpha ,\beta ),d(\beta ,\gamma )\}$ , so, since $\alpha \in D^\beta _{<i}$ and $\beta \in D^\gamma _{<i}$ , we infer that $d(\alpha ,\gamma )<i$ and $\alpha \in D^\gamma _{<i}\cap \beta $ .

$\blacktriangleright $ By Definition 4.1(2), for every $\alpha \in D^\gamma _{<i}\cap \beta $ , $d(\alpha ,\beta )\le \max \{ d(\alpha ,\gamma ),d(\beta ,\gamma )\}$ , so, since $\alpha ,\beta \in D^\gamma _{<i}$ , we infer that $d(\alpha ,\beta )<i$ and $\alpha \in D^\beta _{<i}$ .

Proof. Let d be the function $\rho :[\lambda ^+]^2\rightarrow \lambda $ defined in [Reference Todorcevic45, §9.1]. By [Reference Todorcevic45, Lemma 9.1.1], d is subadditive. By [Reference Todorcevic45, Lemma 9.1.2], d is also locally small (i.e., $|D^\gamma _{\le i}|<\lambda $ for all $\gamma <\lambda ^+$ and $i<\lambda $ ).

Next, by [Reference Rinot and Todorcevic37], we may fix a coloring $c:[\lambda ^+]^2\rightarrow \lambda ^+$ witnessing $\lambda ^+\nrightarrow [\lambda ^+;\lambda ^+]^2_{\lambda ^+}$ . By [Reference Inamdar and Rinot22, Lemma 3.16], this means that for every $A\in [\lambda ^+]^{\lambda ^+}$ , there exists an $\epsilon <\lambda ^+$ such that, for all $\beta \in \lambda ^+\setminus \epsilon $ and $\xi <\epsilon $ , there exists $\alpha \in A\cap \epsilon $ such that $c(\alpha ,\beta )=\xi $ .

We now verify that c and d are as sought.

Proof. Let $\langle M_\gamma \mathrel {|} \gamma <\lambda ^+\rangle $ be a sequence of elementary submodels of $\mathcal H_{\lambda ^{++}}$ , each of size $\lambda $ , such that $\{A,e\}\in M_0$ , such that $M_\gamma \in M_{\gamma +1}$ for every $\gamma <\lambda ^+$ , and such that $M_\delta =\bigcup _{\gamma <\delta }M_\gamma $ for every limit nonzero $\delta <\lambda ^+$ . It follows that $C=\{\gamma <\lambda ^+\mathrel {|} M_\gamma \cap \lambda ^+=\gamma \}$ is a club in $\lambda ^+$ .

We claim that the following club is as sought:

$$ \begin{align*}D=\{\delta<\lambda^+\mathrel{|} \operatorname{\mathrm{otp}}(C\cap\delta)=\lambda^\delta\}.\end{align*} $$

To this end, let $\delta \in D$ , $\beta \in \lambda ^+\setminus \delta $ , $\xi <\delta $ , $i<\lambda $ , and $\eta <\delta $ . We shall find an $\alpha \in A\cap \delta $ above $\eta $ such that $c(\alpha ,\beta )=\xi $ and $d(\alpha ,\beta )>i$ .

For every $\gamma \in C\setminus \xi $ , the set $A_\gamma =A\setminus \gamma $ is in $[\lambda ^+]^{\lambda ^+}\cap M_{\gamma +1}$ , and hence, there exists $\epsilon \in \lambda ^+\cap M_{\gamma +1}$ such that for all $\beta '\in \lambda ^+\setminus \epsilon $ and $\xi '<\epsilon $ , there exists $\alpha '\in A_\gamma \cap \epsilon $ such that $c(\alpha ',\beta ')=\xi '$ . In particular, we may pick $\alpha _\gamma \in A\cap M_{\gamma +1}\setminus \gamma $ such that $c(\alpha _\gamma ,\beta )=\xi $ . It follows that $\gamma \mapsto \alpha _\gamma $ is a strictly increasing function from $C\cap \delta $ to $A\cap \delta $ . As $\delta \in D$ , we infer that $A'=\{ \alpha \in A\cap \delta \mathrel {|} \eta <\alpha \ \&\ c(\alpha ,\beta )=\xi \}$ has size $\lambda $ . As d is locally small, we may now pick $\alpha \in A'\setminus D^\beta _{\le i}$ . Then $\alpha \in A\cap \delta $ above $\eta $ , $d(\alpha ,\beta )>i$ and $c(\alpha ,\beta )=\xi $ , as sought.

This completes the proof.

Proof. We start by verifying a special case.

Proof. For every $\epsilon <\kappa $ , $A\setminus \epsilon $ is in $[\kappa ]^\kappa $ , and as $d:[\kappa ]^2\rightarrow \theta $ witnesses $\operatorname {\mathrm {U}}(\kappa ,2,\theta ,2)$ , it is the case that $d"[A\setminus \epsilon ]^2$ is cofinal in $\theta $ . It thus follows that we may fix a $\kappa $ -sized pairwise disjoint subfamily $\mathcal A$ of $[A]^2$ such that $d(a)>i$ for all $a\in \mathcal A$ . Note that for all $\beta <\kappa $ and $a\in \mathcal A\cap \mathcal P(\beta )$ , there must exist some $\alpha \in a$ such that $d(\alpha ,\beta )>i$ because, by subadditivity,

$$ \begin{align*}i<d(a)\le\max\{d(\min(a),\beta),d(\max(a),\beta)\}.\end{align*} $$

Therefore, it now suffices to prove that there exists some $\gamma <\kappa $ such that for every $\beta \in \kappa \setminus \gamma $ , there exists $a\in \mathcal A\cap \mathcal P(\gamma )$ such that $c[a\times \{\beta \}]=\{\xi \}$ . Towards a contradiction, suppose that this is not the case. For every $\gamma <\kappa $ , fix $\beta _\gamma \in \kappa \setminus \gamma $ such that there exists no $a\in \mathcal A\cap \mathcal P(\gamma )$ with $c[a\times \{\beta _\gamma \}]=\{\xi \}$ . For each $\gamma <\kappa $ , set $a_\gamma =\{\beta _\gamma \}\cup a$ for some $a\in \mathcal A$ such that $\min (a)>\beta _\gamma $ . Fix a club $C\subseteq \kappa $ such that for every $\gamma \in C$ , for every $\bar \gamma <\gamma $ , $\max (a_{\bar \gamma })<\gamma $ . It follows that $\mathcal A'=\{ a_\gamma \mathrel {|} \gamma \in C\}$ is a collection of $\kappa $ -many pairwise disjoint elements of $[\kappa ]^3$ . So, since c witnesses $\operatorname {\mathrm {Pr}}_1(\kappa ,\kappa ,\kappa ,4)$ , we may find $a,b\in \mathcal A'$ with $\max (a)<\min (b)$ such that $c[a\times b]=\{\xi \}$ . Pick $\bar \gamma ,\gamma $ in C such that $a=a_{\bar \gamma }$ and $b=a_\gamma $ . From $\max (a_{\bar \gamma })<\min (a_\gamma )$ , it follows that $\bar \gamma <\gamma $ , and

$$ \begin{align*}\max(a_{\bar\gamma})<\gamma\le\beta_\gamma=\min(a_\gamma).\end{align*} $$

In particular, $a'=a_{\bar \gamma }\setminus \{\beta _\gamma \}$ is an element of $\mathcal A\cap \mathcal P(\gamma )$ and $c[a'\times \{\beta _\gamma \}]=\{\xi \}$ . This is a contradiction.

Now, given $A\in [\kappa ]^\kappa $ , let $\langle M_\gamma \mathrel {|} \gamma <\kappa \rangle $ be a sequence of elementary submodels of $\mathcal H_{\kappa ^+}$ , each of size less than $\kappa $ , such that $\{A,c,d\}\cup \theta \subseteq M_0$ , such that $M_\gamma \in M_{\gamma +1}$ for every $\gamma <\kappa $ , and such that $M_\delta =\bigcup _{\gamma <\delta }M_\gamma $ for every limit nonzero $\delta <\kappa $ . We claim that the following club is as sought:

$$ \begin{align*}D=\{\delta<\kappa\mathrel{|} M_\delta\cap\kappa=\delta\}.\end{align*} $$

To see it, let $\beta \in \kappa \setminus \delta $ , $\xi <\delta $ , $i<\theta $ , and $\epsilon <\delta $ ; we must find $\alpha \in A$ with $\epsilon \le \alpha <\delta $ such that $c(\alpha ,\beta )=\xi $ and $d(\alpha ,\beta )>i$ . The set $A'=A\setminus \epsilon $ is in $[\kappa ]^\kappa \cap M_\delta $ , and so are $\xi $ and i. It thus follows from Claim 4.8.1 that there exists $\gamma \in \kappa \cap M_\delta $ such that for every $\beta '\in \kappa \setminus \gamma $ , there exists $\alpha \in A'\cap \gamma $ such that $c(\alpha ,\beta ')=\xi $ and $d(\alpha ,\beta ')>i$ . As $\gamma <\delta \le \beta $ , it follows that there exists $\alpha \in A'\cap \gamma $ such that $c(\alpha ,\beta )=\xi $ and $d(\alpha ,\beta )>i$ . Evidently, $\epsilon \le \alpha <\delta $ .

Proof. By Theorem 4.4, we may assume that $\theta ^+<\kappa $ . Also, by Theorem 4.8, it suffices to find a map $c:[\kappa ]^2\rightarrow \kappa $ witnessing $\operatorname {\mathrm {Pr}}_1(\kappa ,\kappa ,\kappa ,4)$ , and a subadditive map $d:[\kappa ]^2\rightarrow \theta $ witnessing $\operatorname {\mathrm {U}}(\kappa ,2,\theta ,2)$ .

By [Reference Rinot36, Theorem B], $\square (\kappa )$ implies $\operatorname {\mathrm {Pr}}_1(\kappa ,\kappa ,\kappa ,\theta )$ . Inspecting the proof of [Reference Rinot36, Theorem 3.3] makes it clear that the same conclusion already follows from $\square (\kappa ,{\sqsubseteq _\theta })$ . In addition, by [Reference Lambie-Hanson and Rinot28, Theorem A], $\square (\kappa ,{\sqsubseteq _\theta })$ yields a subadditive witness to $\operatorname {\mathrm {U}}(\kappa ,2,\theta ,2)$ .

Next, by [Reference Lambie-Hanson and Rinot28, Corollary 3.29], the existence of a uniformly coherent $\kappa $ -Souslin tree yields a subadditive witness to $\operatorname {\mathrm {U}}(\kappa ,2,\theta ,2)$ . It is also well known that the existence of a uniformly coherent $\kappa $ -Souslin tree induces a witness to $\operatorname {\mathrm {Pr}}_1(\kappa ,\kappa ,\kappa ,\omega )$ .