# On the unavoidability of oriented trees

## TL;DR

This paper investigates the unavoidable nature of oriented trees within tournaments, establishing bounds on the size of tournaments that must contain any given oriented tree based on its order and number of leaves.

## Contribution

It introduces new bounds on the unavoidable size of tournaments containing arbitrary oriented trees, extending previous results for arborescences.

## Key findings

- Every arborescence of order n with k leaves is (n+k-1)-unavoidable.
- Every oriented tree of order n with k leaves is (3/2 n + 3/2 k - 2)-unavoidable and (9/2 n - 5/2 k - 9/2)-unavoidable.
- Every oriented tree of order n with k leaves is (n + 144k^2 - 280k + 124)-unavoidable.

## Abstract

A digraph is {\it $n$-unavoidable} if it is contained in every tournament of order $n$. We first prove that every arborescence of order $n$ with $k$ leaves is $(n+k-1)$-unavoidable. We then prove that every oriented tree of order $n$ ($n\geq 2$) with $k$ leaves is $(\frac{3}{2}n+\frac{3}{2}k -2)$-unavoidable and $(\frac{9}{2}n -\frac{5}{2}k -\frac{9}{2})$-unavoidable, and thus $(\frac{21}{8} n- \frac{47}{16})$-unavoidable. Finally, we prove that every oriented tree of order $n$ with $k$ leaves is $(n+ 144k^2 - 280k + 124)$-unavoidable.