I have a hopefully quick reference question. Given n vertex digraph G of out-degree T, and no 2 cycles (girth at least 3), what is the lower bound on number of vertices in the largest induced acyclic subgraph of G?

Without the girth constraint, the right answer is n/T (union of a bunch of T cliques). With the girth constraint, I am not even sure T matters, and hope maybe the answer is Omega(n). But anything much more than n/T would be useful.

This must have been studied, but Google is failing me ;).

Thank you in advance!
Yevgeniy