A parametrization of the abstract Ramsey theorem

TL;DR

This paper introduces a parametrization of the abstract Ramsey theorem using perfect subsets of the Cantor space, extending combinatorial forcing techniques and deriving new parametrized results including a version of the Hales-Jewett theorem.

Contribution

It develops a novel parametrization framework for the abstract Ramsey theorem and related combinatorial principles, expanding the applicability of these theorems.

Findings

Parametrization of the abstract Ramsey theorem with perfect subsets.

Extension of combinatorial forcing to the parametrized setting.

A new parametrized version of the Hales-Jewett theorem.

Abstract

We give a parametrization with perfect subsets of $2^{\infty}$ of the abstract Ramsey theorem (see \cite{todo}) Our main tool is an extension of the parametrized version of the combinatorial forcing developed in \cite{nash} and \cite{todo}, used in \cite{mij} to the obtain a parametrization of the abstract Ellentuck theorem. As one of the consequences, we obtain a parametrized version of the Hales-Jewett theorem. Finally, we conclude that the family of perfectly ${\cal S}$-Ramsey subsets of $2^{\infty}\times {\cal R}$ is closed under the Souslin operation. {\bf Key words and phrases}: Ramsey theorem, Ramsey space, parametrization.