Set Theory Seminar - Thomas Gilton (UCLA): Abraham-Rubin-Shelah Open Coloring Axiom with a large continuum

Wed, 01/05/201914:00-15:30
Ross 63

Abstract: In their 1985 paper, the above three authors introduced a consistent generalization of Ramsey's theorem to pairs of countable ordinals, which we abbreviate as $OCA_{ARS}$. This axiom asserts that for any continuous coloring (with respect to an appropriate topology) of pairs of countable ordinals, there is a decomposition of $\omega_1$ into countably-many homogeneous sets. The key to their argument is to construct Preassignments of Colors. However, the known constructions of Preassignments only work over models satisfying the CH; this leads naturally to the question of whether $OCA_{ARS}$ is consistent with a "large continuum," i.e., with $2^{\aleph_0}>\aleph_2$. Recently, Itay Neeman and I have answered this question. The key to our solution is to construct Preassignments with a substantial amount of "symmetry" and to combine them according to a specific recipe, which we refer to as Partition Product. In this talk, we will motivate and define Partition Products, show how to construct symmetric preassignments, and then show how we can use this machinery to obtain our result.