Ramsey Theory for Words over an Infinite Alphabet

TL;DR

This paper develops a comprehensive partition theory for omega-words over infinite alphabets, extending classical combinatorial theorems and providing new results in Ramsey theory for infinite structures.

Contribution

It introduces a novel partition theory for omega-words over infinite alphabets, significantly strengthening existing finite alphabet results and extending key theorems to more general semigroup contexts.

Findings

Extended Hindman, Milliken-Taylor, and van der Waerden theorems to infinite alphabets.

Established strong simultaneous extensions of classical partition theorems.

Provided a unified framework for Ramsey theory over infinite words.

Abstract

A complete partition theory is presented for omega-located words (and omega-words), namely for located words over an infinite alphabet dominated by a fixed increasing sequence. This theory strengthens in an essential way the classical Carlson, Furstenberg-Katznelson, and Bergelson-Blass-Hindman partition theory for words over a finite alphabet. Consequences of this theory are strong simultaneous extensions of the classical Hindman, Milliken-Taylor partition theorem, and of a van der Waerden theorem for general semigroups, extending results of Hindman-Strauss and Beiglbock.