A proof of Sumner's universal tournament conjecture for large tournaments

TL;DR

This paper proves Sumner's universal tournament conjecture for all sufficiently large tournaments, confirming that such tournaments contain any directed tree of a given size, building on previous approximate results.

Contribution

The authors establish the conjecture for large tournaments, advancing the understanding of universal properties in directed graphs.

Findings

Confirmed Sumner's conjecture for large tournaments

Extended previous approximate results to an exact proof

Demonstrated techniques for handling large directed structures

Abstract

Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains any directed tree on $n$ vertices. In this paper we prove that this conjecture holds for all sufficiently large $n$. The proof makes extensive use of results and ideas from a recent paper by the same authors, in which an approximate version of the conjecture was proved.