The Ramsey property implies no mad families.
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- Record sourced from PubMed, PMID 31467168.
- Also identified by DOI 10.1073/pnas.1906183116 and PMC identifier 6754562.
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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, <i>Ann. Math. Logic</i> 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 [Formula: see text] 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.