Bounds on Zimin Word Avoidance

How long can a word be that avoids the unavoidable? Word $W$ encounters word $V$ provided there is a homomorphism $ϕ$ defined by mapping letters to nonempty words such that $ϕ(V)$ is a subword of $W$. Otherwise, $W$ is said to avoid $V$. If, on any arbitrary finite alphabet, there are finitely many words that avoid $V$, then we say $V$ is unavoidable. Zimin (1982) proved that every unavoidable word is encountered by some word $Z_n$, defined by: $Z_1 = x_1$ and $Z_{n+1} = Z_n x_{n+1} Z_n$. Here we explore bounds on how long words can be and still avoid the unavoidable Zimin words.

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