Let $\mathcal{U}$ be (selective) ultrafilter and $\omega = \coprod\_{i<\omega}S\_i$ be partition of $\omega$ with small sets $S\_i\notin\mathcal{U}$. All $S\_i$ are infinite. Does there exist a system of bijections $\varphi\_i:\omega\to S\_i$ such that for any big set $B\in\mathcal{U}$ and any system of big sets $\{B\_i\}\_{i\in B}$ the set $\cup\_{i\in B}\varphi\_i(B\_i)$ is big?



On infinite combinatorics of ultrafilters
Like your previous question, [Selective ultrafilter and bijective mapping](https://mathoverflow.net/questions/324254) , this fails for all nonprincipal ultrafilters $\mathcal U$ on $\omega$, and for essentially the same reason. If there were such bijections $\phi\_i$, then the function $f:\omega\to\omega$ that is constant on each $S\_i$ with value $i$ would satisfy $f(\mathcal U)=\mathcal U$, yet it is not equal to the identity (or even one-to-one) on any set in $\mathcal U$. That contradicts the theorem that I quoted in my answer to your previous question (and for which Ali Enayat kindly provided a reference).


