# Abraham-Rubin-Shelah Open Colorings and a Large Continuum

## TL;DR

This paper proves the consistency of the Abraham-Rubin-Shelah Open Coloring Axiom with a continuum of size , resolving a long-standing open question by developing new symmetry techniques and a novel poset called a Partition Product.

## Contribution

It introduces a method to construct names for color preassignments over models with CH using symmetry, enabling the consistency proof with a large continuum.

## Key findings

- Established the consistency of the Open Coloring Axiom with continuum 
- Developed symmetry-based techniques for constructing color names
- Introduced the Partition Product poset for combining models

## Abstract

We show that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with $2^{\aleph_0}=\aleph_3$. This answers one of the main open questions from the 1985 paper of Abraham-Rubin-Shelah. As in their paper, we need to construct names for so-called preassignments of colors in order to add the necessary homogeneous sets. However, these names are constructed over models satisfying the CH. In order to address this difficulty, we show how to construct such names with very strong symmetry conditions. This symmetry allows us to combine them in many different ways, using a new type of poset called a Partition Product, and thereby obtain a model of this axiom in which $2^{\aleph_0}=\aleph_3$.