Ramsey Theory for Words Representing Rationals

TL;DR

This paper develops a systematic Ramsey theory for words over infinite alphabets, extending previous work, and applies it to derive partition theorems for rational numbers and semigroups, advancing combinatorial mathematics.

Contribution

It extends Carlson's Ramsey theory to countable ordinals and Schreier families for words over infinite alphabets, and applies this to rational numbers and semigroups.

Findings

Established a Ramsey theory for {}-Z*-located words over infinite alphabets.

Derived a partition theorem for rational numbers.

Obtained stronger partition theorems for arbitrary semigroups.

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.