We study the notion of J-MAD families where J is a
Borel ideal on ω. We show that if J is an arbitrary
Fσ ideal, or is any finite or countably iterated Fubini product of
Fσ ideals, then there are no analytic infinite J-MAD
families, and assuming Projective Determinacy there are no infinite projective
J-MAD families; and under the full Axiom of Determinacy +
V=L(R) there are no infinite J-mad families.
These results apply in particular when J is the ideal of finite sets
Fin, which corresponds to the classical notion of MAD families. The
proofs combine ideas from invariant descriptive set theory and forcing.Comment: 40 page