  We show that Ramsey theory, a domain presently conceived to guarantee the
existence of large homogeneous sets for partitions on k-tuples of words (for
every natural number k) over a finite alphabet, can be extended to one for
partitions on Schreier-type sets of words (of every countable ordinal). Indeed,
we establish an extension of the partition theorem of Carlson about words and
of the (more general) partition theorem of Furstenberg-Katznelson about
combinatorial subspaces of the set of words (generating from k-tuples of words
for any fixed natural number k) into a partition theorem about combinatorial
subspaces (generating from Schreier-type sets of words of order any fixed
countable ordinal). Furthermore, as a result we obtain a strengthening of
Carlson's infinitary Nash-Williams type (and Ellentuck type) partition theorem
about infinite sequences of variable words into a theorem, in which an infinite
sequence of variable words and a binary partition of all the finite sequences
of words, one of whose components is, in addition, a tree, are assumed,
concluding that all the Schreier-type finite reductions of an infinite
reduction of the given sequence have a behavior determined by the
Cantor-Bendixson ordinal index of the tree-component of the partition, falling
in the tree-component above that index and in its complement below it.
