A Note on Powers of Paths in Tournaments

TL;DR

This paper improves bounds on the existence of powers of directed paths within tournaments, showing a nearly optimal inverse exponential relation with respect to path length and path power.

Contribution

It establishes a significantly better lower bound for the length of paths whose powers are contained in any tournament, refining previous exponential bounds.

Findings

Every tournament on n vertices contains the k-th power of a directed path of length at least n/2^{6k+7}.

The bound improves upon previous results by Scott and Korándi.

The result aligns with Yuster's upper bound, indicating near-optimality.

Abstract

In this note we show that every tournament on $n$ vertices contains the $k$-th power of a directed path of length $n/2^{6k+7}$, which improves upon the recent bound of Scott and Kor\'{a}ndi of $n/2^{2^{3k}}$. By doing so, we get an inverse exponential dependence on $k$, which is best possible as Yuster recently showed an upper bound of $kn/{2^{k/2}}$.