The Ramsey property implies no mad families
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- OA-The Ramsey property implies no mad families
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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 |
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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 | |
Status | Udgivet - 2019 |
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