Abraham–Rubin–Shelah open colorings and a large continuum

Journal of Mathematical Logic 22 (1) (2022)
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Abstract

We show that the Abraham–Rubin–Shelah Open Coloring Axiom is consistent with a large continuum, in particular, consistent with [Formula: see text]. This answers one of the main open questions from [U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of [Formula: see text]-dense real order types, Ann. Pure Appl. Logic 325(29) (1985) 123–206]. As in [U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of [Formula: see text]-dense real order types, Ann. Pure Appl. Logic 325(29) (1985) 123–206], we need to construct names for the so-called preassignments of colors in order to add the necessary homogeneous sets. However, the known constructions of preassignments (ours in particular) only work assuming the [Formula: see text]. 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. Partition products may be thought of as a restricted memory iteration with stringent isomorphism and coherent-overlap conditions on the memories. We finally construct, in [Formula: see text], the partition product which gives us a model of [Formula: see text] in which [Formula: see text].

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