The Ramsey property implies no mad families

Research output: Contribution to journalJournal articleResearchpeer-review


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 languageEnglish
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number38
Pages (from-to)18883-18887
Number of pages5
Publication statusPublished - 2019

    Research areas

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

Number of downloads are based on statistics from Google Scholar and

No data available

ID: 229102855