Let $D$ be a digraph. Its acyclic number $\vec{\alpha}(D)$ is the maximum order of an acyclic induced subdigraph and its dichromatic number $\vec{\chi}(D)$ is the least integer $k$ such that $V(D)$ can be partitioned into $k$ subsets inducing acyclic subdigraphs. We study ${\vec a}(n)$ and $\vec t(n)$ which are the minimum of $\vec\alpha(D)$ and the maximum of $\vec{\chi}(D)$, respectively, over all oriented triangle-free graphs of order $n$. For every $\epsilon>0$ and $n$ large enough, we show $(1/\sqrt{2} - \epsilon) \sqrt{n\log n} \leq \vec{a}(n) \leq \frac{107}{8} \sqrt n \log n$ and $\frac{8}{107} \sqrt n/\log n \leq \vec{t}(n) \leq (\sqrt 2 + \epsilon) \sqrt{n/\log n}$. We also construct an oriented triangle-free graph on 25 vertices with dichromatic number~3, and show that every oriented triangle-free graph of order at most 17 has dichromatic number at most 2.

翻译：设 $D$ 为有向图。其无环数 $\vec{\alpha}(D)$ 是最大无环导出子有向图的阶数，其二分色数 $\vec{\chi}(D)$ 是使得 $V(D)$ 可划分为 $k$ 个诱导无环子有向图的子集的最小整数 $k$。本文研究所有 $n$ 阶无向三角自由有向图上的 $\vec\alpha(D)$ 最小值 ${\vec a}(n)$ 和 $\vec{\chi}(D)$ 最大值 $\vec t(n)$。对于每个 $\epsilon>0$ 且 $n$ 足够大时，我们证明 $(1/\sqrt{2} - \epsilon) \sqrt{n\log n} \leq \vec{a}(n) \leq \frac{107}{8} \sqrt n \log n$ 以及 $\frac{8}{107} \sqrt n/\log n \leq \vec{t}(n) \leq (\sqrt 2 + \epsilon) \sqrt{n/\log n}$。此外，我们构造了一个 25 个顶点的无向三角自由有向图，其二分色数为 3，并证明任意阶数不超过 17 的无向三角自由有向图其二分色数至多为 2。