Application of entropy compression in pattern avoidance

TL;DR

This paper proves that certain patterns with multiple variables are avoidable over small alphabets, improving bounds on pattern avoidability in combinatorics on words.

Contribution

It provides a positive answer to a longstanding problem, establishing new bounds for pattern avoidability over 2- and 3-letter alphabets.

Findings

Patterns with k variables of length at least 2^k are 3-avoidable.

Patterns with k variables of length at least 3×2^{k-1} are 2-avoidable.

Improves previous bounds by Bell, Goh, and Rampersad.

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 $f= (p)$ 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 improves previous bounds due to Bell and Goh, and Rampersad.