  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.
