Optimality of the minimum semidegree threshold for antidirected-tree containment

Determine whether the minimum semidegree bound \(\delta^0(D)\geq (\ell/(2\ell-1)+\gamma)k\) in the paper’s theorem for balanced antidirected trees is essentially best possible for every integer \(\ell\geq 3\).

Background

The main theorem proves that, for sufficiently large $n$, every $n$-vertex digraph satisfying sufficiently large maximum in- and outdegree and minimum semidegree at least $(\ell/(2\ell-1)+\gamma)k$ contains every balanced antidirected tree with $k$ arcs and polylogarithmically bounded maximum degree.

The authors explain that the threshold is asymptotically sharp when $\ell=2$, because undirected extremal examples can be viewed as digraphs. They explicitly leave unresolved whether the corresponding threshold is essentially sharp for $\ell\geq3$.