Avoidability of formulas with two variables

TL;DR

This paper investigates the avoidability of certain word patterns with at most two variables, determining their avoidability over binary alphabets and the abundance of avoiding words.

Contribution

It classifies the avoidability of formulas with up to two variables over binary alphabets and analyzes the number of words avoiding these patterns.

Findings

Identified which formulas with two variables are 2-avoidable.

Established whether these formulas are avoided by exponentially many binary words.

Abstract

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables 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 consider the patterns such that at most two variables appear at least twice, or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is $2$-avoidable, and if it is $2$-avoidable, we determine whether it is avoided by exponentially many binary words.