An approximate version of Sumner's universal tournament conjecture

TL;DR

This paper proves an approximate version of Sumner's universal tournament conjecture, showing large enough tournaments contain all directed trees of a certain size, with improved bounds for trees of bounded degree.

Contribution

It establishes asymptotic bounds for the conjecture, including the best possible bounds for trees with bounded maximum degree.

Findings

Tournaments on (2+o(1))n vertices contain all directed trees on n vertices.

Tournaments on (1+o(1))n vertices contain all bounded degree directed trees on n vertices.

The results are asymptotically optimal for bounded degree trees.

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 $\Delta$.