• Corpus ID: 227239220

# Paths of given length in tournaments.

```@article{Sah2020PathsOG,
title={Paths of given length in tournaments.},
author={Ashwin Sah and Mehtaab Sawhney and Yufei Zhao},
journal={arXiv: Combinatorics},
year={2020}
}```
• Published 1 December 2020
• Mathematics
• arXiv: Combinatorics
What is the maximum possible number of directed k-edge paths in an n-vertex tournament? We are interested in the regime of fixed k and large n. The expected number of directed k-edge paths in a uniform random n-vertex tournament is n(n− 1) · · · (n− k)/2k = (1+ o(1))n(n/2)k . In this short note we show that one cannot do better, thereby confirming an unpublished conjecture of Jacob Fox, Hao Huang, and Choongbum Lee. Theorem 1. Every n-vertex tournament has at most n(n/2)k directed k-edge paths.

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