Higher-dimensional Delta-systems

TL;DR

This paper extends the classical Delta-system lemma to higher dimensions, focusing on systems of ordinal sets, and applies these results to forcing and partition relations on the reals.

Contribution

It introduces a higher-dimensional version of the Delta-system lemma and demonstrates its applications in set theory, particularly in forcing and partition relations.

Findings

Established a higher-dimensional Delta-system lemma.

Ensured order-theoretic uniformities for systems of ordinal sets.

Applied the lemma to problems involving forcing and partition relations.

Abstract

We investigate higher-dimensional $\Delta$-systems indexed by finite sets of ordinals, isolating a particular definition thereof and proving a higher-dimensional version of the classical $\Delta$-system lemma. We focus in particular on systems that consist of sets of ordinals, in which case useful order-theoretic uniformities can be ensured. We then present three applications of these higher-dimensional $\Delta$-systems to problems involving the interplay between forcing and partition relations on the reals.