On some variations of coloring problems of infinite words

TL;DR

This paper explores advanced coloring problems of infinite words, introducing stronger monochromatic factorization concepts and linking them to key results in Ramsey theory, with probabilistic existence results for ultra monochromatic factorizations.

Contribution

It introduces new notions of monochromatic factorizations for infinite words and connects these to fundamental Ramsey theory theorems, providing probabilistic existence results.

Findings

Existence of ultra monochromatic factorizations for almost all infinite words under Bernoulli measure.

Links between factorization properties and Ramsey theory results like Hindman's theorem.

Almost all words in the subshift have ultra monochromatic factorizations under finite colorings.

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 $A^\omega.$