The Open Coloring Axiom

TL;DR

This paper explains the Open Coloring Axiom, explores its implications in set theory and topology, and proves its consistency with ZFC, showing it is independent of CH and can be assumed in set-theoretic frameworks.

Contribution

It provides a detailed explanation of the Open Coloring Axiom, demonstrates its consistency with ZFC, and explores its implications in topology and infinitary combinatorics.

Findings

The axiom implies =2, conflicting with CH.

The axiom is consistent with ZFC.

It has applications in topology and set theory.

Abstract

This work is concerned with an axiom introduced by Todorc\v{e}vi\'{c} in \cite{stevo} that constitutes a Ramsey-like statement regarding the topology of the reals. Our aim is to explain the axiom in detail, give some interesting applications and finally prove that the axiom is indeed consistent with ZFC, so that it makes sense to consider working with it in the first place. For this particular academic endeavor, we cover several advanced topics in set theory, including concepts like {\sl Hausdorff gaps}, forcing, infinitary combinatorics and a tad of topology. We employ, for example, an argument based on Rothberger's theorem to show that the Open Coloring Axiom implies the equality $\mathfrak b=\aleph_2$, which in turn makes this axiom inconsistent with CH. In other words, in ZFC, the Open Coloring Axiom could be false. To prove its relative consistency, we show that the axiom could be true by following a rather long and technical lemma of Todorc\v{e}vi\'{c}, which leads to the culmination of this work.