leaf set, but for any such collection with a fixed number $m$ of trees with $m\geq 8$.

Let ${{\mathcal P}}=\{{{\mathcal T}}_1,{{\mathcal T}}_2,\ldots,{{\mathcal T}}_m\}$ be a collection of rooted binary phylogenetic $X$-trees. [CPS-Existence]{} is [NP]{}-complete for any fixed $m$ with $m\geq 8$.

Clearly, [CPS-Existence]{} for $m= t+8$ with $t \geq 0$ is in NP. To establish the corollary, we show how one can modify the reduction that is described prior to Theorem \[lem:CPS-Existence\] to obtain a set of $t+8$ rooted binary phylogenetic trees from an instance of [4-Disjoint-Intermezzo]{}. Let $I$ be an instance of [4-Disjoint-Intermezzo]{}. Throughout the remainder of the proof, we assume that there exists an $1 \leq \ell \leq 4$ such that $|B_\ell \cup C_\ell| > t$. Otherwise, since $t = m - 8$ and $m$ is fixed, it follows that $I$ has a constant number $c$ of pairs and triples with $c\leq 4(m-8)$ and is solvable in polynomial time.

Now, let $B_i$ and $C_i$ with $i \in \{1,\,2,\,3,\,4\}$ be a collection of pairs and triples, respectively, such that $|B_i \cup C_i| > t$. Theorem \[lem:CPS-Existence\] establishes the result for when $t=0$. We may therefore assume that $t>0$ and consider two cases. First, suppose that $t$ is even. Replace 