Semidegree threshold for spanning trees in oriented graphs

Abstract

We show that for all $\gamma >0$ and $\Delta \in \mathbb{N}$, there is some $n_0$ such that, if $n\ge n_0$, then every oriented graph on $n$ vertices with minimum semidegree at least $(3/8+\gamma )n$ contains a copy of each oriented tree on $n$ vertices with maximum degree at most $\Delta$. This is asymptotically best possible.

40 pages, 4 figures; v2 updates Question 1.2 and its discussion, and adds a new figure