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 a⃗(n) and t⃗(n) which are the minimum of α⃗(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.