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For a rational number $r$ such that $1<r\leq 2$, an undirected $r$-power is aword of the form $xyx'$, where the word $x$ is nonempty, the word $x'$ is in$\{x,x^R\}$, and we have $|xyx'|/|xy|=r$. The undirected repetition thresholdfor $k$ letters, denoted $\mbox{URT}(k)$, is the infimum of the set of all $r$such that undirected $r$-powers are avoidable on $k$ letters. We firstdemonstrate that $\mbox{URT}(3)=\tfrac{7}{4}$. Then we show that$\mbox{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that$\mbox{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm thisconjecture for $k\in\{4,5,\ldots,21\}.$ We then consider related problems inpattern avoidance; in particular, we find the undirected avoidability index ofevery binary pattern. This is an extended version of a paper presented at WORDS2019, and it contains new and improved results.

The undirected repetition threshold and undirected pattern avoidance