Lengths of extremal square-free ternary words

Abstract

A square-free word w over a fixed alphabet Σ is extremal if every word obtained from w by inserting a single letter from Σ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal square-free word, and demonstrated that there are arbitrarily long extremal square-free ternary words. We find all lengths which admit an extremal square-free ternary word. In particular, we show that there is an extremal square-free ternary word of every sufficiently large length. We also solve the analogous problem for circular words.

Extremal Pattern-Avoiding Words

“…In 2020, Mol and Rampersad [7] adapted the ideas of Grytczuk et al to find all integers n for which an extremal square-free ternary word of length n exists. Mol, Rampersad, and Shallit then studied extremal overlap-free binary words in [6], where an overlap is a word of the form aXaXa, for any letter a and any (possibly empty) word X.…”