A note on the Canonical Ramsey Theorem and Ramsey ultrafilters

TL;DR

This paper characterizes Ramsey ultrafilters on the power set of natural numbers using functions and their ultrafilter extensions, linking partition homogeneity to canonical forms.

Contribution

It provides a new characterization of Ramsey ultrafilters via functions and ultrafilter extensions, connecting partition homogeneity to canonical partitions.

Findings

Existence of a finite partition of $[]^{2n}$ for any partition of $[]^n$

Homogeneous sets for the finite partition are finite unions of canonical sets

New insights into the structure of Ramsey ultrafilters and their relation to partition properties

Abstract

We give a characterizations of Ramsey ultrafilters on $\mathscr P(\omega)$ in terms of functions $f:\omega^n\to\omega$ and their ultrafilter extensions. To do this, we prove that for any partition $\mathcal P$ of $[\omega]^n$ there is a finite partition~$\mathcal Q$ of $[\omega]^{2n}$ such that any set $X\subseteq \omega$ that is homogeneous for $\mathcal Q$ is a finite union of sets that are canonical for $\mathcal P$.