Unavoidable trees in tournaments

TL;DR

This paper establishes conditions under which certain oriented trees are unavoidable in tournaments, proving most are unavoidable and improving bounds for embedding trees with polylogarithmic maximum degree.

Contribution

It provides a sufficient condition for an oriented tree to be unavoidable and confirms that almost all labeled oriented trees are unavoidable, also improving embedding bounds.

Findings

Almost all labeled oriented trees are unavoidable.

Every tournament on n + o(n) vertices contains all trees with polylogarithmic maximum degree.

A new sufficient condition for unavoidability of oriented trees.

Abstract

An oriented tree $T$ on $n$ vertices is unavoidable if every tournament on $n$ vertices contains a copy of $T$. In this paper we give a sufficient condition for $T$ to be unavoidable, and use this to prove that almost all labelled oriented trees are unavoidable, verifying a conjecture of Bender and Wormald. We additionally prove that every tournament on $n + o(n)$ vertices contains a copy of every oriented tree $T$ on $n$ vertices with polylogarithmic maximum degree, improving a result of K\"uhn, Mycroft and Osthus.