  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 $A^\omega.$
