Spanning trees in dense directed graphs

In 2001, Komlos, Sarkozy and Szemeredi proved that, for each alpha > 0, there is some c > 0 and n(0) such that, if n >= n(0), then every n-vertex graph with minimum degree at least (1/2 +alpha)n contains a copy of every n-vertex tree with maximum degree at most cn/ log n. We prove the corresponding result for directed graphs. That is, for each alpha > 0, there is some c > 0 and n0 such that, if n >= n(0), then every n-vertex directed graph with minimum semi-degree at least (1/2 + alpha)n contains a copy of every n-vertex oriented tree whose underlying maximum degree is at most cn/log n. As with Komlos, Sarkozy and Szemeredi's theorem, this is tight up to the value of c. Our result improves a recent result of Mycroft and Naia, which requires the oriented trees to have underlying maximum degree at most delta, for any constant delta is an element of N and sufficiently large n. In contrast to these results, our methods do not use Szemeredi's regularity lemma. (C) 2022 Elsevier Inc.