Fragile and Robust Squarefree Infinite Words

Definitions and mathematical statement

Let $A=\{0,1,2\}$. The source does not specify whether infinite words are one-sided or two-sided. Accordingly, let $I$ denote either $\{0,1,2,\ldots\}$ or ${\mathbb{Z}}$, with the intended choice unresolved. A word $w=(w_i)_{i\in I}\in A^I$ is squarefree if it contains no contiguous factor $uu$, where $u$ is a nonempty finite word. Equivalently, there are no $i\in I$ and integer $m\geq1$, with $i,i+1,\ldots,i+2m-1\in I$, such that \[ w_{i+j}=w_{i+m+j}\qquad(0\leq j<m). \] For a nonempty finite set $F\subset I$ and a replacement assignment $a\in\prod_{i\in F}(A\setminus\{w_i\})$, define the modified word \[ w^{F,a}_i=\begin{cases}a_i,&i\in F,\\w_i,&i\notin F.\end{cases} \] This interprets a symbol flip as replacing a letter by either of the other two letters; the source does not specify a ternary flip operation.

Under this interpretation, a squarefree word $w$ is fragile if \[ \forall\,\varnothing\ne F\subset I\text{ finite},\quad \forall\,a\in\prod_{i\in F}(A\setminus\{w_i\}),\quad w^{F,a}\text{ is not squarefree}. \] A candidate precise interpretation of robust is that there exist nonempty finite sets $F_1,F_2,\ldots\subset I$ and assignments $a^{(k)}\in\prod_{i\in F_k}(A\setminus\{w_i\})$ such that \[ F_k\cap F_\ell=\varnothing\quad(k\ne\ell),\qquad w^{F_k,a^{(k)}}\text{ is squarefree for every }k\geq1. \] Here each modification is applied separately to the original word. Whether every replacement assignment must work, or whether combinations of these modifications must also preserve squarefreeness, is not specified.

Provisional existence questions. For the intended index set and flip convention, does there exist an infinite ternary squarefree fragile word? Separately, does there exist an infinite ternary squarefree robust word? These are provisional separate questions: the source's phrase "such words that are robust" could instead require the same word to be both fragile and robust, which is incompatible with the displayed definitions. The intended relationship between the two questions requires clarification.

Short English statement

The record asks about infinite squarefree words on three letters whose squarefreeness is destroyed by every finite nontrivial change, and about words admitting infinitely many changes on disjoint finite sets of positions that preserve squarefreeness. The indexing, meaning of a ternary flip, replacement quantifiers, and relationship between the two existence questions remain unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001558$, source problem code catalogue entry, title Fragile and Robust Squarefree Infinite Words, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2010.
• Supplied source citation: Mini-Workshop: Combinatorics on Words (2011), Oberwolfach Reports, DOI 10.4171/owr/2010/37, pp. 2195--2244. Original workshop report: https://doi.org/10.4171/owr/2010/37. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions. The citation's year 2011 differs from the recorded proposed year and the year appearing in the DOI; this discrepancy is unresolved.
• Supplied literature reference: Przybocki, Lengths of Irreducible and Delicate Words, https://arxiv.org/abs/2108.06646.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem partially solved and states that robust ternary squarefree words are established by Brinkhuis-type substitutions, while the stronger infinite fragile-word construction remains unresolved and later work treats only finite or related delicacy notions. This is supplied dated curation, not an independently verified literature assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying verification describes only enclosing TeX notation in math delimiters without changing mathematical content. It does not establish mathematical correctness or resolve the formulation issues above.