In this paper we consider the following question in the spirit of Ramsey
theory: Given x∈Aω, 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 Aω. 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∈Aω satisfying λx​(n+1)−λx​(n)=1
for all n sufficiently large, where λx​(n) denotes the number of
distinct factors of x of length n.Comment: arXiv admin note: incorporates 1301.526