Every orientation of a 4-chromatic graph has a non-bipartite acyclic subgraph

Let f(n) denote the smallest integer such that every directed graph with chromatic number larger than f(n) contains an acyclic subgraph with chromatic number larger than n. The problem of bounding this function was introduced by Addario-Berry et al., who noted that f(n) ≤ n². The only improvement over this bound was obtained by Nassar and Yuster, who proved that f(2) = 3 using a deep theorem of Thomassen. Yuster asked if this result can be proved using elementary methods. In this short note we provide such a proof.