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Schreier sets in Ramsey theory


Authors: V. Farmaki and S. Negrepontis
Journal: Trans. Amer. Math. Soc. 360 (2008), 849-880
MSC (2000): Primary 05D10
DOI: https://doi.org/10.1090/S0002-9947-07-04323-1
Published electronically: September 24, 2007
MathSciNet review: 2346474
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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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Additional Information

V. Farmaki
Affiliation: Department of Mathematics, Athens University, Panepistemiopolis, Athens 157 84, Greece
Email: vfarmaki@math.uoa.gr

S. Negrepontis
Affiliation: Department of Mathematics, Athens University, Panepistemiopolis, Athens 157 84, Greece
Email: snegrep@math.uoa.gr

DOI: https://doi.org/10.1090/S0002-9947-07-04323-1
Keywords: Ramsey theory, Schreier sets, words
Received by editor(s): October 23, 2005
Published electronically: September 24, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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