Coloring Problems for Infinite Words

Abstract

Given a finite coloring (or finite partition) of the free semigroup ${\mathcal A}^+$ over a set ${\mathcal A}$, we consider various types of monochromatic factorizations of right-sided infinite words $x\in {\mathcal A}^\omega$. In 2006 T. Brown asked the following question in the spirit of Ramsey theory: Given a nonperiodic infinite word x = x₁x₂x₃⋯ with values in a set ${\mathcal A}$, does there exist a finite Coloring $\varphi : {\mathcal A} ^+\rightarrow C$ relative to which x does not admit a φ-monochromatic factorization, i.e., a factorization of the form x = u₁u₂u₃⋯ with φ(u _i ) = φ(u _j ) for all i, j ≥ 1? We give an optimal affirmative answer to this question by showing that if x = x₁x₂x₃⋯ is an infinite nonperiodic word with values in a set ${\mathcal A},$ then there exists a 2-coloring $\varphi : {\mathcal A} ^+\rightarrow \{0,1\}$ such that for any factorization x = u₁u₂u₃⋯, we have φ(u _i ) ≠ φ(u _j ) for some i ≠ j. Some stronger versions of the usual notion of monochromatic factorization are also introduced and studied. We establish links, and in some cases equivalences, between the existence of these factorizations and fundamental results in Ramsey theory including the infinite Ramsey theorem, Hindman’s finite sums theorem, partition regularity of IP-sets, and the Milliken–Taylor theorem.