# The undirected repetition threshold

For rational 1<r≤ 2, an undirected r-power is a word of the form xyx', where x≠ε, x'∈{x,x^R}, and |xyx'|/|xy|=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)=74. Then we show that URT(k)≥k-1k-2 for all k≥ 4. We conjecture that URT(k)=k-1k-2 for all k≥ 4, and we confirm this conjecture for k∈{4,8,12}.

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