A Coloring Problem for Infinite Words

TL;DR

This paper investigates a Ramsey-theoretic coloring problem for infinite words, proving that almost all words and certain classes of recurrent words admit a finite coloring of factors avoiding monochromatic factorizations.

Contribution

It establishes that for almost all infinite words and specific recurrent classes, a finite coloring prevents monochromatic factorizations, extending Ramsey theory to infinite words.

Findings

Almost all infinite words can be finitely colored to avoid monochromatic factorizations.

Certain classes of recurrent words, including aperiodic balanced words, satisfy the coloring property.

Words with linear factor complexity also admit such colorings.

Abstract

In this paper we consider the following question in the spirit of Ramsey theory: Given $x\in A^\omega,$ where $A$ is a finite non-empty set, does there exist a finite coloring of the non-empty factors of $x$ with the property that no factorization of $x$ is monochromatic? We prove that this question has a positive answer using two colors for almost all words relative to the standard Bernoulli measure on $A^\omega.$ We also show that it has a positive answer for various classes of uniformly recurrent words, including all aperiodic balanced words, and all words $x\in A^\omega$ satisfying $\lambda_x(n+1)-\lambda_x(n)=1$ for all $n$ sufficiently large, where $ \lambda_x(n)$ denotes the number of distinct factors of $x$ of length $n.$