An approximate version of Sumnerʼs universal tournament conjecture

Sumnerʼs universal tournament conjecture states that any tournament on 2 n − 2 vertices contains a copy of any directed tree on n vertices. We prove an asymptotic version of this conjecture, namely that any tournament on ( 2 + o ( 1 ) ) n vertices contains a copy of any directed tree on n vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed Δ, any tournament on ( 1 + o ( 1 ) ) n vertices contains a copy of any directed tree on n vertices with maximum degree at most Δ.