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arXiv · 1904.05824

The Ramsey property implies no mad families

David Schrittesser·Asger Törnquist
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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. This solves a long-standing problem going back to Mathias \cite{mathias}. 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.

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David Schrittesser, Asger Törnquist. 2019-04-11. The Ramsey property implies no mad families. https://doi.org/10.1073/pnas.1906183116

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