We say that a digraph $D$ is *complete $r$-partite* when $G(D)$ is complete $r$-partite. We say that $D$ is *monotone* when $V(D)$ can be partitioned into subsets $V_1, V_2, \ldots, V_r$ such that every arc in $V_i \times V_j$ satisfies $i < j$. We cite a result given in [@IPL2013] as a lemma:

[@IPL2013] \[LEMITA\] Let $D$ be a complete $r$-partite digraph. Then, $\eta_t(D) < +\infty$ if and only $D$ is monotone. In that case, $$\eta_t(D) = \max \biggl\{ \biggl\lceil \dfrac{s_i}{|V_i|} \biggr\rceil : i \in [r] \biggr\},$$ where $V_1, \ldots, V_r$ is the partition of $V(D)$, $s_r = |V_r|$ and $s_i = \max\{1 + s_{i+1}, |V_i|\}$ for all $i \in [r-1]$.

Let $G=(V_1 \cup \cdots \cup V_r,E)$ be the complete $r$-partite graph ($V_1$,$\ldots$, $V_r$ are its stable sets) and $|V_i| \geq |V_{i+1}|$ for all $i \in [r-1]$. Then, $\eta(G) = \max \{ \lceil \frac{s_i}{|V_i|} \rceil : i \in [r] \}$ where $s_r = |V_r|$ and $s_i = \max\{1 + s_{i+1}, |V_i|\}$ for all $i \in [r-1]$. Moreover, $\eta(G) \leq r$.

Let $D$ be the monotone digraph such that $G(D) = G$ and the partition of $V(D)$ is $V_1, V_2, \ldots, V_r$. We must prove that $D$ represents the acyclic orientation 