Mad families and ultrafilters

Abstract

For each almost disjoint family X let F ( X ) = { a ⊆ ω : card { s ∈ X : s ∖ a is finite } = 2 ω } , I ( X ) = { a ⊆ ω : card { s ∈ X : card ( s ∩ a ) = ω } = 2 ω } 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 ω ) 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 F(X) = I(Y) = p .