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](https://mathoverflow.net/questions/325925/the-property-of-the-dense-subfilter-of-a-selective-ultrafilter) which still has no answer.



Dense subfilter of selective ultrafilter
The answer is negative. The equivalent question was answered [here](https://mathoverflow.net/questions/325925/the-property-of-the-dense-subfilter-of-a-selective-ultrafilter) by Andreas Blass


