An approximate version of Sumnerʼs universal tournament conjecture

Sumner’s universal tournament conjecture states that any tournament on 2n−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 ∆, any tournament on (1 + o(1))n vertices contains a copy of any directed tree on n vertices with maximum degree at most ∆.