The undirected repetition threshold and undirected pattern avoidance

For a rational number $r$ such that $1<r\le 2$, an undirected $r$-power is a word of the form $xy{x}^{\mathrm{\prime }}$, where the word $x$ is nonempty, the word $x^{\mathrm{\prime }}$ is in $\{x,x^R\}$, and we have $\mathrm{\mid }xy{x}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{/}\mathrm{\mid }xy\mathrm{\mid }=r$. The undirected repetition threshold for $k$ letters, denoted $URT(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $URT(3)=\frac{7}{4}$. Then we show that $URT(k)\ge \frac{k-1}{k-2}$ for all $k\ge 4$. We conjecture that $URT(k)=\frac{k-1}{k-2}$ for all $k\ge 4$, and we confirm this conjecture for $k\in \{4,5,\dots ,21\}\mathrm{.}$ We then consider related problems in pattern avoidance; in particular, we find the undirected avoidability index of every binary pattern. This is an extended version of a paper presented at WORDS 2019, and it contains new and improved results.