# 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.