Acyclic subgraphs of digraphs with high chromatic number

Abstract

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 $f(G) \ge n^{5/9-o(1)}s^{-14/9}$ 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 $f(G) \ge n^{5/9-o(1)}$ which polynomially improves the folklore bound $f(G) \ge n^{1/2-o(1)}$. 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 $f(G) \ge n^{5/9-o(1)}$. 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.