  We prove that every $n$-vertex tournament $G$ has an acyclic subgraph with
chromatic number at least $n^{5/9-o(1)}$, while there exists an $n$-vertex
tournament $G$ whose every acyclic subgraph has chromatic number at most
$n^{3/4+o(1)}$. This establishes in a strong form a conjecture of Nassar and
Yuster and improves on another result of theirs. Our proof combines
probabilistic and spectral techniques together with some additional ideas. In
particular, we prove a lemma showing that every tournament with many transitive
subtournaments has a large subtournament that is almost transitive. This may be
of independent interest.
