# The undirected repetition threshold

## TL;DR

This paper introduces the undirected repetition threshold, determines it exactly for three letters, establishes a lower bound for more letters, and conjectures its exact value for all larger alphabet sizes, confirming it for specific cases.

## Contribution

The paper defines the undirected repetition threshold, proves its value for three letters, provides a lower bound for larger alphabets, and conjectures its exact value, confirming it for some specific cases.

## Key findings

- URT(3)=7/4
- URT(k)≥(k-1)/(k-2) for k≥4
- Conjecture URT(k)=(k-1)/(k-2) for all k≥4, confirmed for k=4,8,12

## Abstract

For rational $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx'$, where $x$ is nonempty, $x'\in\{x,x^\mathrm{R}\}$, and $|xyx'|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mathrm{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $\mathrm{URT}(3)=\tfrac{7}{4}$. Then we show that $\mathrm{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that $\mathrm{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm this conjecture for $k\in\{4,8,12\}.$