Polish space partition principles and the Halpern-L\"auchli theorem

TL;DR

This paper explores partition principles related to Polish space products that provide simple proofs of the Halpern-L"auchli theorem and investigates their consistency and limitations within set theory.

Contribution

It isolates combinatorial principles underlying the Halpern-L"auchli theorem and demonstrates their consistency via forcing, also establishing their non-provability in ZFC.

Findings

Partition principles yield straightforward proofs of the Halpern-L"auchli theorem.

Forcing can establish the consistency of these principles.

These principles impose lower bounds on the continuum size.

Abstract

The Halpern-L\"auchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern-L\"auchli theorem, and the same forcing from Harrington's proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the size of the continuum.