Infinite ternary square-free words concatenated from permutations of a single word

TL;DR

This paper proves the existence of infinite square-free ternary words with specific factorizations and morphisms, resolving longstanding questions in combinatorics on words.

Contribution

It demonstrates the existence of uniform ternary morphisms of length at least 23 and infinite square-free words with n-stem factorizations for all n ≥ 13.

Findings

Existence of infinite square-free ternary words with n-stem factorizations for all n ≥ 13

Existence of uniform ternary morphisms of length k for every k ≥ 23

Almost complete resolution of a problem posed by the author and Rampersad

Abstract

We answer a question of Harju: An infinite square-free ternary word with an $n$-stem factorization exists for any $n\ge 13$. We show that there are uniform ternary morphisms of length $k$ for every $k\ge 23$. This resolves almost completely a problem of the author and Rampersad.