{
  "id": "math/0510102",
  "version": "v1",
  "published": "2005-10-05T17:11:49.000Z",
  "updated": "2005-10-05T17:11:49.000Z",
  "title": "Schreier Sets in Ramsey Theory",
  "authors": [
    "V. Farmaki",
    "S. Negrepontis"
  ],
  "categories": [
    "math.CO"
  ],
  "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 (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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-10-05T17:11:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "ramsey theory",
      "schreier sets",
      "partition theorem",
      "infinite sequence",
      "combinatorial subspaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10102F"
    }
  }
}