The Open Coloring Axiom
Tonatiuh Matos-Wiederhold
TL;DR
The Open Coloring Axiom (OCA) is a Ramsey-type principle for the real line with strong consequences, including $\mathfrak b=\aleph_2$ under OCA and thus incompatibility with CH. The paper surveys OCA, develops Todorcevic's Lemma as a central technical tool, and builds a forcing framework to establish the relative consistency of OCA with ZFC. It then links OCA to PFA, showing that PFA implies OCA, thereby deriving the consistency strength of OCA from large-cardinal–assisted forcing. Together, these results illuminate deep connections between infinitary combinatorics, topology, and forcing in the landscape of set theory.
Abstract
This work is concerned with an axiom introduced by Todorcěvić 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 Todorcěvić, which leads to the culmination of this work.