Antidirected forests in digraphs

A digraph is antidirected if every vertex has indegree zero or outdegree zero. Let $k\ge 2$, and let $F$ be an antidirected forest with $k$ arcs and no isolated vertices. We prove that every digraph $D$ of order $n$ with more than $g_k(n):=2\max \{\left(\genfrac{}{}{0px}{}{2k-1}{2}\right),(k-1)(n-\frac{k}{2})\}$ arcs contains $F$ as a subdigraph. For $n\ge 2k-1$, this threshold equals $2\mathrm{e}\mathrm{x}(n,k{K}_2)$ and is attained by symmetric digraphs arising from extremal $k{K}_2$-free graphs. Consequently, the maximum directed extremal number over all such forests is $2\mathrm{e}\mathrm{x}(n,k{K}_2)$. The proof combines a counting inequality for rooted antidirected forests, embeddings extending vertex-disjoint arcs, and vertex deletion. In the remaining case, the Gallai–Edmonds decomposition of the underlying graph gives the required bound on the number of arcs.