Suppose every set of real numbers has the Ramsey property and
“uniformization on Ellentuck-comeager sets” as well as Dependent Choice
hold (as is the case under the Axiom of Determinacy, but also in
Solovay’s model). Then there are no MAD families. As it turns out, there
are also no (Fin x Fin)-MAD families, where Fin x Fin is the
two-dimensional Fubini product of the ideal of finite sets. We also
comment on higher dimensional products.
Results are joint work with Asger Törnquist.