The Ramsey property implies no mad families

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The Ramsey property implies no mad families. / Schrittesser, David; Törnquist, Asger.

In: Proceedings of the National Academy of Sciences of the United States of America, Vol. 116, No. 38, 2019, p. 18883-18887.

Research output: Contribution to journalJournal articlepeer-review

Harvard

Schrittesser, D & Törnquist, A 2019, 'The Ramsey property implies no mad families', Proceedings of the National Academy of Sciences of the United States of America, vol. 116, no. 38, pp. 18883-18887. https://doi.org/10.1073/pnas.1906183116

APA

Schrittesser, D., & Törnquist, A. (2019). The Ramsey property implies no mad families. Proceedings of the National Academy of Sciences of the United States of America, 116(38), 18883-18887. https://doi.org/10.1073/pnas.1906183116

Vancouver

Schrittesser D, Törnquist A. The Ramsey property implies no mad families. Proceedings of the National Academy of Sciences of the United States of America. 2019;116(38):18883-18887. https://doi.org/10.1073/pnas.1906183116

Author

Schrittesser, David ; Törnquist, Asger. / The Ramsey property implies no mad families. In: Proceedings of the National Academy of Sciences of the United States of America. 2019 ; Vol. 116, No. 38. pp. 18883-18887.

Bibtex

@article{08356b0aabdb446b934974b8a8640d97,
title = "The Ramsey property implies no mad families",
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.",
keywords = "Borel ideals, Invariant descriptive set theory, Maximal almost disjoint families, Ramsey property",
author = "David Schrittesser and Asger T{\"o}rnquist",
year = "2019",
doi = "10.1073/pnas.1906183116",
language = "English",
volume = "116",
pages = "18883--18887",
journal = "Proceedings of the National Academy of Sciences of the United States of America",
issn = "0027-8424",
publisher = "The National Academy of Sciences of the United States of America",
number = "38",

}

RIS

TY - JOUR

T1 - The Ramsey property implies no mad families

AU - Schrittesser, David

AU - Törnquist, Asger

PY - 2019

Y1 - 2019

N2 - 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.

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.

KW - Borel ideals

KW - Invariant descriptive set theory

KW - Maximal almost disjoint families

KW - Ramsey property

UR - http://www.scopus.com/inward/record.url?scp=85072315429&partnerID=8YFLogxK

U2 - 10.1073/pnas.1906183116

DO - 10.1073/pnas.1906183116

M3 - Journal article

C2 - 31467168

AN - SCOPUS:85072315429

VL - 116

SP - 18883

EP - 18887

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 38

ER -

ID: 229102855