The undirected repetition threshold and undirected pattern avoidance

For a rational number r such that 1<r≤2, an undirected r-power is word of the form xyx′, where x nonempty, x′ in {x,xR}, and we have |xyx′|/|xy|=r. The repetition threshold for k letters, denoted URT(k), infimum set all r-powers are avoidable on letters. We first demonstrate URT(3)=74. Then show URT(k)≥k−1k−2 k≥4. conjecture URT(k)=k−1k−2 k≥4, confirm this k∈{4,5,…,21}. then consider related problems pattern avoidance; particular, find avoidability index every binary pattern. This extended version paper presented at WORDS 2019, it contains new improved results.