The Ramsey property and higher dimensional mad families

We prove, using only weak choice principles, that if every set of reals has the Ramsey Property, then there are no infinite maximal almost disjoint families with respect to the transfinitely iterated Fr\'{e}chet ideals. These results were announced by the authors in the Proceedings of the National Academy of Sciences of the U.S.A. Complementing the above, we also show that the same conclusion cannot be obtained in the pointclass $\Sigma^1_2$ if the Ramsey Property is replaced by the assumption that all $\Sigma^1_2$ sets are Laver measurable.Comment: 42 pages. The paper has been thoroughly revised and the exposition has been improved from start to finish. Typos and errors, many of which were highly confusing, have been corrected throughou