A family of formulas with reversal of high avoidability index

We present an infinite family of formulas with reversal whose avoidability index is bounded between 4 and 5, and we show that several members of the family have avoidability index 5. This family is particularly interesting due to its size and the simple structure of its members. For each k ∈ {4,5}, there are several previously known avoidable formulas (without reversal) of avoidability index k, but they are small in number and they all have rather complex structure.http://dx.doi.org/10.1142/S021819671750024

The Weak Circular Repetition Threshold Over Large Alphabets

The repetition threshold for words on $n$ letters, denoted \mbox{RT}(n), is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetition thresholds for $n$ letters, denoted \mbox{CRT}_{\mbox{W}}(n), \mbox{CRT}_{\mbox{I}}(n), and \mbox{CRT}_{\mbox{S}}(n), respectively. Currie and the present authors conjectured that \mbox{CRT}_{\mbox{I}}(n)=\mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n) for all $n\geq 4$. We prove that \mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n) for all $n\geq 45$, which confirms a weak version of this conjecture for all but finitely many values of $n$.Comment: arXiv admin note: text overlap with arXiv:1911.0577