Open AccessAvoidability of Formulas with Two Variables
Author(s) -
Pascal Ochem,
Matthieu Rosenfeld
Publication year - 2017
Publication title -
the electronic journal of combinatorics/the journal of combinatorics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.703
H-Index - 52
eISSN - 1097-1440
pISSN - 1077-8926
DOI - 10.37236/6536
Subject(s) - mathematics , morphism , alphabet , combinatorics , combinatorics on words , word (group theory) , sigma , binary number , integer (computer science) , discrete mathematics , arithmetic , linguistics , philosophy , physics , geometry , quantum mechanics , computer science , programming language
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.