The Ramsey property implies no mad families

David Schrittesser; Asger Törnquist

doi:10.1073/pnas.1906183116

The Ramsey property implies no mad families

David Schrittesser, Asger Törnquist^*

^*Corresponding author for this work

Department of Mathematical Sciences

2 Citations (Scopus)

8 Downloads (Pure)

Abstract

We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E₀ and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.

Original language	English
Journal	Proceedings of the National Academy of Sciences of the United States of America
Volume	116
Issue number	38
Pages (from-to)	18883-18887
Number of pages	5
ISSN	0027-8424
DOIs	https://doi.org/10.1073/pnas.1906183116
Publication status	Published - 2019

Keywords

Borel ideals
Invariant descriptive set theory
Maximal almost disjoint families
Ramsey property

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OA-The Ramsey property implies no mad familiesFinal published version, 703 KBLicence: CC BY-NC-ND

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AB - We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0 and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.

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The Ramsey property implies no mad families

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