Schreier sets in Ramsey theory

Abstract

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 CantorBendixson ordinal index of the tree-component of the partition, falling in the tree-component above that index and in its complement below it.

The Strength of Ramsey’s Theorem for Coloring Relatively Large Sets

“…In fact, Pudlák-Rödl [20] prove a generalization of the above theorem to any countable ordinal (see infra for more details). As observed in [9], Schreier families turn out to essentially coincide with the concept of exactly large set. The classical Schreier family is defined as the following set {s = {n 1 , .…”

“…First, we conjecture that our results generalize to the transfinite generalizations above ω of the notions of large set, Schreier family, and Turing jump. The notions of α-large set, α-Schreier family, and α-Turing jump are all well-defined and studied for every countable ordinal (see, respectively, [13], [9], and [27] for definitions). As mentioned in the introduction, RT(!ω) generalizes nicely to colorings of α-Schreier families, or, equivalently, of exactly α-large sets.…”

A Dichotomy Principle for Universal Series

“…Extensions involving Schreier sets have been given by Farmaki (2004) [F], and by Farmaki-Negrepontis (2006) [FN1], (2008) [FN2]. These results, or others of the same type, have found important applications in various branches of mathematics, notably in Banach space theory; we refer the reader to the survey paper by Gowers (2003) [G].…”

New bounds on the strength of some restrictions of Hindman’s Theorem

“…3 Note that ACA 0 is known to be equivalent to RT 3 2 (Ramsey's Theorem for 2-colorings of triples) by seminal work of Jockusch and of Simpson ([32,Theorem III.7.6] or [22,Chapter 6]), so we have that HT implies RT 3 2 over RCA 0 . On the other hand ACA + 0 was only recently given a Ramsey-theoretic characterization in work of the first and fourth author, who showed [8] that the system ACA + 0 is equivalent to a Ramsey-theoretic theorem due to Pudlák and Rödl [31] and Farmaki and Negrepontis [16], which we denote by RT !ω 2 (see Definition 5.4). This theorem extends Ramsey's Theorem to colourings of objects of variable dimension, in particular to so-called exactly large sets of integers, where a set is exactly large in case its cardinality is greater by one than its minimum element.…”