The Ramsey property and higher dimensional mad families

David Schrittesser, Asger Törnquist

Published 2020-03-24Version 1

Suppose every set has the Ramsey property and Ramsey-co-null uniformization, as well as the Principle of Dependent Choice hold. Then there is no infinite $\mathcal I$-mad family, for any ideal $\mathcal I$ in smallest class of ideals containing the Fr\^echet ideal and closed under taking Fubini sums. In fact, we show a local form of this theorem which in turn has many consequences, improving and unifying the proofs of several results which were already known for classical mad families. These results were previously announced in Proceedings of the National Academy of Sciences of the U.S.A. We show as a contrasting result that there is a co-analytic infinite mad family in the Laver extension of $L$.