Schreier sets in Ramsey theory

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 ( generated from k-tuples of words for any fixed natural number k) into a partition theorem about combinatorial subspaces ( generated 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 in finite 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.