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We prove L67, using its definitions of $W_3$, $R_1$ and the pattern coloring. The pattern is

$$A_3,B_3,A_1,B_3,A_2,B_3,A_1,B_3,A_3. \tag{1}$$

Cliques of arbitrary finite size

Observation. For every integer $n\ge2$, the coloring just defined has a color-$1$ clique of size $n$. Its trees can have strictly increasing root lengths.

Here is an explicit construction. Put $M=2n(n+2)$, and for $1\le i\le n$ and $1\le k\le4$ put

$$\ell_{ik}=(k-1)M+i, \qquad r_{ik}=kM-(n+2)i.$$

Represent a tree by an ordered tuple of its root children, each a tuple of its label-$1$ children, each of those a tuple of leaf labels. Define

$$T_i=\bigl\langle \langle\langle\ell_{i1},r_{i1}\rangle\rangle, \langle\langle\ell_{i2},r_{i2}\rangle\rangle, \langle\langle\ell_{i3}\rangle,\langle r_{i3}\rangle\rangle, \langle\langle\ell_{i4},r_{i4}\rangle\rangle, \langle\langle r_{i4}+1\rangle\rangle,\ldots, \langle\langle r_{i4}+i\rangle\rangle \bigr\rangle. \tag{2}$$

All leaves increase from left to right, and every branch has height three. Thus $T_i\in W_3$. It has $4+i$ root children, so $T_1<\cdots<T_n$ in the required well-order.

Fix $i<j$. In each of the first three bands the order is

$$\ell_{ik}<\ell_{jk}<r_{jk}<r_{ik}.$$

The fourth band has the same nesting, with the extra leaves inside their respective right-hand ends:

$$\ell_{i4}<\ell_{j4}<r_{j4}<r_{j4}+1<\cdots<r_{j4}+j <r_{i4}<r_{i4}+1<\cdots<r_{i4}+i.$$

Indeed $r_{j4}+j<r_{i4}$ follows from $(n+2)(j-i)-j\ge2$. Also $\ell_{nk}<r_{nk}$, and every band ends before the next begins. These inequalities show that all the trees' leaf sets are disjoint.

The five cells of $T_i$ relative to $T_j$ are

$$\{\ell_{i1}\},\quad \{r_{i1},\ell_{i2}\},\quad \{r_{i2},\ell_{i3}\},\quad \{r_{i3},\ell_{i4}\},\quad \{r_{i4},r_{i4}+1,\ldots,r_{i4}+i\}.$$

Their levels are $3,1,2,1,3$. For the second and fourth cells, the largest leaf's label-$1$ ancestor contains its right partner outside the cell. For the middle cell, the label-$1$ ancestor of $\ell_{i3}$ is a singleton inside the cell, while its label-$2$ ancestor also contains $r_{i3}$ outside the cell. The endpoint levels follow the rule.

The four cells of $T_j$ are its first three full root subtrees and the union of its fourth root subtree with all its extra singleton subtrees. The two interior cells have level $3$: each contains the full root subtree of its largest leaf, but not the whole tree. The first and last cells also have level $3$ by definition. Interleaving the two lists gives exactly (1). Since this holds for every $i<j$, the trees in (2) form the claimed clique.

Degeneracy on good partitions

Let $D$ be a convex partition of $\operatorname{Seq}(T)$ for $T\in W_3$ such that the last child of every nonterminal node is outside $G(T,D)$. We show that every interior cell has $R_1$ level $1$.

To see this, let $E$ be an interior cell and let $x$ be the leaf labeled by $\max E$. Since $E$ is not the last cell, Definition 9 puts $x$ in $X(T,D)$ and hence in $G(T,D)$. Let $y$ be its parent, of ordinal label $1$. The assumed condition says that the last child of $y$ is not in $G(T,D)$, so $x$ is not that last child. A later child of $y$ is a leaf with label strictly larger than $\max E$. It lies outside $E$. Thus $y$ is the lowest ancestor of $x$ whose leaf set is not contained in $E$, giving $R_1(E)=1$. Good partitions satisfy the assumed condition by Definition 11(3), so this conclusion applies to them.

In particular, the $A_2$ interior cell in (1) cannot occur when the induced partition of $A$ is good. The pattern also requires interior $B_3$ cells, which cannot occur when the induced partition of $B$ is good. Restricting the candidate coloring to the paper's good pairs therefore removes every occurrence of this pattern. It cannot supply the required pattern realization on such pairs.