An approximate version of Sumner's universal tournament conjecture

Abstract

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 $\Delta$, any tournament on $(1+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices with maximum degree at most $\Delta$.