Schreier sets in Ramsey theory

Farmaki, V.; Negrepontis, S.

doi:10.1090/S0002-9947-07-04323-1

Schreier sets in Ramsey theory
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Trans. Amer. Math. Soc. 360 (2008), 849-880 Request permission

Abstract:

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 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.

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