Acyclic subgraphs of digraphs with high chromatic number

For a digraph G , let f ( G ) be the maximum chromatic number of an acyclic subgraph of G . For an n -vertex digraph G it is proved that f ( G ) ≥ n 5 / 9 − o ( 1 ) s − 14 / 9 where s is the bipartite independence number of G , i.e., the largest s for which there are two disjoint s -sets of vertices with no edge between them. This generalizes a result of Fox, Kwan and Sudakov, who proved this for the case s = 0 (i.e., tournaments and semicomplete digraphs). Consequently, if s = n o ( 1 ) , then f ( G ) ≥ n 5 / 9 − o ( 1 ) which polynomially improves the folklore bound f ( G ) ≥ n 1 / 2 − o ( 1 ) . As a corollary, with high probability, all orientations of the random n -vertex graph with edge probability p = n − o ( 1 ) (in particular, constant p , hence almost all n -vertex graphs) satisfy f ( G ) ≥ n 5 / 9 − o ( 1 ) . Our proof uses a theorem of Gallai and Milgram that together with several additional ideas, essentially reduces to the proof of Fox, Kwan and Sudakov.