Ramsey and Nash-Williams combinatorics via Schreier families

TL;DR

This paper extends classical combinatorial theorems like Ramsey and Nash-Williams using Schreier families, providing stronger results and new proofs for Ellentuck's and Galvin-Prikry's theorems.

Contribution

It introduces an extension of the finite Ramsey theorem and employs Schreier families to unify and strengthen classical partition theorems.

Findings

Extended the finite Ramsey partition theorem.

Derived a stronger form of the infinite Nash-Williams partition theorem.

Provided a new proof of Ellentuck's and Galvin-Prikry's theorems.

Abstract

The main results of this paper (a) extend the finite Ramsey partition theorem, and (b) employ this extension to obtain a stronger form of the infinite Nash-Williams partition theorem, and also a new proof of Ellentuck's, and hence Galvin-Prikry's partition theorem. The proper tool for this unification of the classical partition theorems at a more general and stronger level is the system of Schreier families $({\cal A}_{\xi})$ of finite subsets of the set of natural numbers, defined for every countable ordinal $\xi$.