Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order

Let D be a digraph. Its acyclic number (D) is the maximum order of an acyclic induced subdigraph and its dichromatic number (D) is the least integer k such that V (D) can be partitioned into k subsets inducing acyclic subdigraphs. We study a (n) and t (n) which are the minimum of (D) and the maximum of (D), respectively, over all oriented triangle-free graphs of order n. For every >0 and n large enough, we show (1/2 -) n n a (n) 1078 n n and 8107 n/ n t (n) (2 +) n/ 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.