Unavoidable trees in tournaments

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.