Mad families and ultrafilters

Abstract:

For each almost disjoint family X let $F(X) = \{ a \subseteq \omega :{\text {card}}\{ s \in X:s\backslash a\;{\text {is}}\;{\text {finite}}\} = {2^\omega }\} ,I(X) = \{ a \subseteq \omega :{\text {card}}\;\{ s \in X:{\text {card}}\;(s \cap a) = \omega \} = {2^\omega }\}$ . Assuming $P({2^\omega })$ we show that for each nonprincipal ultrafilter p there exist a maximal almost disjoint family X and an almost disjoint family Y with $F(X) = I(Y) = p$.