Antidirected subgraphs of oriented graphs

We show that for every $\eta>0$ every sufficiently large $n$-vertex oriented graph D of minimum semidegree exceeding $(1 + \eta) k/2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k \ge \eta n$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs. Further, we show that in the same setting, D contains every $k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length 1 or 2 span a forest. As a special case, we can find all antidirected cycles of length at most $k$. Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed and Thomass\'e for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$-vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$.