A coloring problem for infinite words

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