Parameterized partition relations on the real numbers

Archive for Mathematical Logic 48 (2):201-226 (2009)


We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition of ${\mathbb{R}^\mathbb{N}}$ there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of ${[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}$ there is ${X \in [\mathbb{N}]^{\mathbb{N}}}$ and a sequence ${\{P_{k} : k \in \mathbb{N}\}}$ of perfect sets such that the product ${[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}$ lies in one piece of the partition, where ${[X]^{\mathbb{N}}}$ is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness

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References found in this work

Happy Families.A. R. D. Mathias - 1977 - Annals of Mathematical Logic 12 (1):59.
Infinite Combinatorics and Definability.Arnold W. Miller - 1989 - Annals of Pure and Applied Logic 41 (2):179-203.
Solovay Models and Forcing Extensions.Joan Bagaria & Roger Bosch - 2004 - Journal of Symbolic Logic 69 (3):742-766.
Regularity Properties of Definable Sets of Reals.Jacques Stern - 1985 - Annals of Pure and Applied Logic 29 (3):289-324.
Proper Forcing Extensions and Solovay Models.Joan Bagaria & Roger Bosch - 2004 - Archive for Mathematical Logic 43 (6):739-750.

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Citations of this work

Absoluteness Via Resurrection.Giorgio Audrito & Matteo Viale - 2017 - Journal of Mathematical Logic 17 (2):1750005.
Happy and Mad Families in L.Itay Neeman & Zach Norwood - 2018 - Journal of Symbolic Logic 83 (2):572-597.
On Coding Uncountable Sets by Reals.Joan Bagaria & Vladimir Kanovei - 2010 - Mathematical Logic Quarterly 56 (4):409-424.

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