Blow-Ups and Extensions of Trees in Tournaments

A class of acyclic digraphs $\mathcal{C}$ is linearly unavoidable if there exists a constant $c$ such that every digraph $D\in \mathcal{C}$ is contained in all tournaments of order $c\cdot |V(D)|$. The class of all acyclic digraphs is not linearly unavoidable, and Fox, He, and Wigderson recently showed that this is not even the case for acyclic digraphs with bounded maximum degree. On the positive side, Häggkvist and Thomason proved that the class of oriented trees is linearly unavoidable. In this work, we generalize this result to acyclic digraphs obtained from an oriented tree by adding at most $k$ vertices, and $k$-blow-ups of oriented trees, for every fixed integer $k$. More precisely, we show that if $D$ is obtained from an oriented tree $F$ of sufficiently large order $n$ by adding $k$ universal vertices, then $D$ is contained in all tournaments on $2\cdot 3^{(k+1)(2k+1)} \cdot n$ vertices; and if $D$ is obtained from $F$ by replacing each vertex $u$ by an independent set $X_u$ of size $k$ and every arc $uv$ by all possible arcs from $X_u$ to $X_v$, then $D$ is contained in every tournament on $2^{10+18k}k \cdot n$ vertices.