In 2001, Komlós, Sárközy and Szemerédi proved that, for each $\alpha >0$, there is some $c>0$ and $n_0$ such that, if $n\ge n_0$, 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 $n_0$ such that, if $n\ge n_0$, 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 Komlós, Sárközy and Szemerédi'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 Δ, for any constant $\mathrm{\Delta }\in \mathbb{N}$ and sufficiently large n. In contrast to these results, our methods do not use Szemerédi's regularity lemma.

2001 年，Komlós、Sárközy 和 Szemerédi 证明，对于每个$\alpha >0$， 有一些$C>0$和$n_0$这样，如果$n\ge n_0$, 则至少每个度数最小的n顶点图$(1/2+\alpha )n$包含最多具有最大度数的每个n顶点树的副本$Cn/\mathrm{日志}n$. 我们证明了有向图的相应结果。也就是说，对于每个$\alpha >0$， 有一些$C>0$和$n_0$这样，如果$n\ge n_0$, 则每一个至少具有最小半度数的n顶点有向图$(1/2+\alpha )n$包含每个面向n顶点的树的副本，其基础最大度数最多为$Cn/\mathrm{日志}n$.

与 Komlós、Sárközy 和 Szemerédi 定理一样，这与c的值紧密相关。我们的结果改进了 Mycroft 和 Naia 的最近结果，这要求定向树具有最多 Δ 的潜在最大度数，对于任何常数$\mathrm{\Delta }\in \mathbb{ñ}$并且足够大的n。与这些结果相反，我们的方法不使用 Szemerédi 的正则引理。