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Given selective ultrafilter $\mathcal{U}$ on $\omega$ and dense filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$, where $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ if the limit exists. Let $\mathcal{F}=\mathcal{F_1}\cap\mathcal{U}$.

Question: Does there exist a family $\{A_i\subset\omega\}_{i<\omega},~A_i=\{a_{ik}\}_{k<\omega}$ of pairwise disjoint subsets such that for any $B\in\mathcal{F}$ we have: $$ \{a_{ik}~|~i,k\in B\}\in\mathcal{U} $$

Remark: The question is equivalent formulation of this one which still has no answer.

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The answer is negative. The equivalent question was answered here by Andreas Blass

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