The Undirected Repetition Threshold And Undirected Pattern Avoidance

For a rational number r such that 1 < r <= 2, an undirected r power is a word of the form xyx', where the word x is nonempty, the word x' is in {x, x(R)}, and we have vertical bar xyx'vertical bar/vertical bar xy vertical bar = 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) = 7/4. Then we show that URT(k) >= k-1/k-2 for all k >= 4. We conjecture that URT(k) = k-1/k-2 for all k >= 4, and we confirm this conjecture for k is an element of {4,5, ..., 21}. We then consider k-2 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. (C) 2021 Elsevier B.V. All rights reserved.