  An acyclic set in a digraph is a set of vertices that induces an acyclic
subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$
vertices without directed 2-cycles possesses an acyclic set of size at least
$3n/5$. We prove this conjecture for digraphs where every directed cycle has
length at least 8. More generally, if $g$ is the length of the shortest
directed cycle, we show that there exists an acyclic set of size at least $(1 -
3/g)n$.
