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 af dette arbejde

Institut for Matematiske Fag

2 Citationer (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.

Originalsprog	Engelsk
Tidsskrift	Proceedings of the National Academy of Sciences of the United States of America
Vol/bind	116
Udgave nummer	38
Sider (fra-til)	18883-18887
Antal sider	5
ISSN	0027-8424
DOI	https://doi.org/10.1073/pnas.1906183116
Status	Udgivet - 2019

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10.1073/pnas.1906183116

OA-The Ramsey property implies no mad familiesForlagets udgivne version, 703 KBLicens: CC BY-NC-ND

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Citationsformater

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AU - Törnquist, Asger

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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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