A Coloring Problem for Sturmian and Episturmian Words

TL;DR

This paper proves that for certain classes of infinite words, specifically Sturmian and standard episturmian words, there always exists a finite coloring of factors preventing monochromatic factorizations, addressing a Ramsey theory-inspired open question.

Contribution

It establishes the existence of such colorings for Sturmian and standard episturmian words, advancing understanding in combinatorics on words and Ramsey theory.

Findings

Existence of finite colorings for Sturmian words

Existence of finite colorings for standard episturmian words

Addresses an open Ramsey theory question

Abstract

We consider the following open question in the spirit of Ramsey theory: Given an aperiodic infinite word $w$, does there exist a finite coloring of its factors such that no factorization of $w$ is monochromatic? We show that such a coloring always exists whenever $w$ is a Sturmian word or a standard episturmian word.