The Ramsey property implies no mad families

Abstract

We show that if all collections of infinite subsets of $\mathbb{N}$ 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.

10 pages; fixed a mistake in Theorem 4.4

References in corpus (4)

• Can you take Toernquist's inaccessible away?
• Compactness of maximal eventually different families
• Maximal almost disjoint families, determinacy, and forcing
• Madness and regularity properties