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arXiv · 1307.2828

A Coloring Problem for Infinite Words

Aldo de Luca·Elena V. Pribavkina·Luca Q. Zamboni
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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 $n.$

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Aldo de Luca, Elena V. Pribavkina, Luca Q. Zamboni. 2013-07-10. A Coloring Problem for Infinite Words. https://arxiv.org/abs/1307.2828

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