Ramsey Theory for Words Representing Rationals
Abstract
Ramsey theory for words over a finite alphabet was unified in the work of Carlson and Furstenberg-Katznelson. Carlson, in the same work, outlined a method to extend the theory for words over an infinite alphabet, but subject to a fixed dominating principle, proving in particular an Ellentuck version, and a corresponding Ramsey theorem for k=1. In the present work we develop in a systematic way a Ramsey theory for words (in fact for {\omega}-Z*-located words) over a doubly infinite alphabet extending Carlson's approach (to countable ordinals and Schreier-type families), and we apply this theory, exploiting the Budak-Isik-Pym representation, to obtain a partition theory for the set of rational numbers. Furthermore, we show that the theory can be used to obtain partition theorems for arbitrary semigroups, stronger than known ones.
Cite
@article{arxiv.1011.0580,
title = {Ramsey Theory for Words Representing Rationals},
author = {Vassiliki Farmaki and Andreas Koutsogiannis},
journal= {arXiv preprint arXiv:1011.0580},
year = {2010}
}