# Subdivisions of digraphs in tournaments

## TL;DR

This paper proves that high out-degree tournaments contain subdivisions of complete directed graphs and determines bounds for the smallest tournament size guaranteeing a transitive subdivision, extending classical graph theory results.

## Contribution

It establishes the minimum out-degree condition for subdivisions of complete digraphs in tournaments and bounds the size of tournaments needed for transitive subdivisions, advancing directed graph theory.

## Key findings

- High out-degree ensures subdivisions of complete digraphs
- Bounds on tournament size for transitive subdivisions
- Extension of classical clique subdivision theorems

## Abstract

We show that for every positive integer $k$, any tournament with minimum out-degree at least $(2+o(1))k^2$ contains a subdivision of the complete directed graph on $k$ vertices, which is best possible up to a factor of $8$. This may be viewed as a directed analogue of a theorem proved by Bollob\'as and Thomason, and independently by Koml\'os and Szemer\'edi, concerning subdivisions of cliques in graphs with sufficiently high average degree. We also consider the following problem: given $k$, what is the smallest positive integer $f(k)$ such that any $f(k)$-vertex tournament contains a $1$-subdivision of the transitive tournament on $k$ vertices? We show that $f(k)= O\left (k^2\log^3 k\right)$ which is best possible up to the logarithmic factors.