On some variations of coloring problems of infinite words

Abstract

Given a finite coloring (or finite partition) of the free semigroup $A^+$ over a set $A$, we consider various types of monochromatic factorizations of right sided infinite words $x\in A^\omega$. Some stronger versions of the usual notion of monochromatic factorization are introduced. A factorization is called sequentially monochromatic when concatenations of consecutive blocks are monochromatic. A sequentially monochromatic factorization is called ultra monochromatic if any concatenation of arbitrary permuted blocks of the factorization has the same color of the single blocks. 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. We prove that for each finite set $A$ and each finite coloring $\varphi: A^+\rightarrow C,$ for almost all words $x\in A^\omega,$ there exists $y$ in the subshift generated by $x$ admitting a $\varphi$-ultra monochromatic factorization, where "almost all" refers to the Bernoulli measure on $A^\omega.$