Theory of sets – Selective ultrafilter dense subfilter

Selective ultrafilter die $ mathcal {U} $ in $ omega $ and dense filter $ mathcal {F_1} = {A subset omega ~ | ~ rho (A) = 1 } $, where $ rho (A) = lim_ {n a infty} frac {| A cap n |} {n} $ If the limit exists Leave $ mathcal {F} = mathcal {F_1} cap mathcal {U} $.

Question: Is there a family? $ {A_i subset omega } _ {i < omega}, ~ A_i = {a_ {ik} } _ {k < omega} $ of disjoint subsets by pairs such that for any $ B en mathcal {F} $ we have:
$$
{a_ {ik} ~ | ~ i, ​​k in B } in mathcal {U}
$$

Observation: The question is the equivalent formulation of this that still has no answer.