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

Let D be a digraph. Its acyclic number α⃗(D) is the maximum order of an acyclic induced subdigraph and its dichromatic number χ⃗(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 α(D) and the maximum of χ⃗(D), respectively, over all oriented triangle-free graphs of order n. For every ε>0 and n large enough, we show (1/√2 - ε) √nlog n ≤ a⃗(n) ≤ 107/8 √ n log n and 8/107 √ n/log n ≤ t⃗(n) ≤ (√ 2 + ε) √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.