Abstract
We characterize the formulas that are avoided by every $\alpha$-free word for some $\alpha>1$. We show that the avoidable formulas whose fragments are of the form $XY$ or $XYX$ are $4$-avoidable. The largest avoidability index of an avoidable palindrome pattern is known to be at least $4$ and at most $16$. We make progress toward the conjecture that every avoidable palindrome pattern is $4$-avoidable.
Highlights
IntroductionThe avoidability index λ(p) of a pattern p is the size of the smallest alphabet Σ such that there exists an infinite word over Σ containing no occurrence of p
A pattern p is a non-empty finite word over an alphabet ∆ = {A, B, C, . . .} of capital letters called variables
The avoidability index λ(p) of a pattern p is the size of the smallest alphabet Σ such that there exists an infinite word over Σ containing no occurrence of p
Summary
The avoidability index λ(p) of a pattern p is the size of the smallest alphabet Σ such that there exists an infinite word over Σ containing no occurrence of p. An occurrence of a formula f in a word w is a non-erasing morphism h : ∆∗ → Σ∗ such that the h-image of every fragment of f is a factor of w. As for patterns, the index λ(f ) of a formula f is the size of the smallest alphabet allowing the existence of an infinite word containing no occurrence of f. A formula f is nice if for every variable X of f , there exists a fragment of f that contains X at least twice.
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