The undirected repetition threshold

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 $k\in\{4,8,12\}.$