Abstract
In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f= h(p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We give a positive answer to Problem 3.3.2 in Lothaire's book "Algebraic combinatorics on words'", that is, every pattern with $k$ variables of length at least $2^k$ (resp. $3\times2^{k-1}$) is 3-avoidable (resp. 2-avoidable). This conjecture was first stated by Cassaigne in his thesis in 1994. This improves previous bounds due to Bell and Goh, and Rampersad.
Highlights
A pattern p is a non-empty 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 w over Σ containing no occurrence of p
We obtain a contradiction by showing that the number of possible outputs is strictly smaller than the number of possible inputs when t is chosen large enough compared to n. This implies that every pattern p with at most k variables and length at least q(k) is 2-avoidable
Summary
We consider upper bounds on the avoidability index of long enough patterns with k variables. } have length 3 × 2v(p)−1 − 1 and are not 2-avoidable This shows that the upper bound 3 of Theorem 2.(a) is best possible. The avoidability index of every pattern with at most 3 variables is known, thanks to various results in the literature. For v(p) = 2, every binary pattern of length at least 4 contains a square, and is 3-avoidable. Roth [13] proved that every binary pattern of length at least 6 is 2-avoidable. We prove Theorem 2.(a) in Section 3 as a corollary of a result of Bell and Goh [2]. Blanchet-Sadri and Woodhouse [4] independently proved Theorem 2 using completely different methods
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