## 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 E_{0} 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 |
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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 | |

Publication status | Published - 2019 |

## Keywords

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