On infinite combinatorics of ultrafilters

Question

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?

Answer 1

Like your previous question, Selective ultrafilter and bijective mapping , 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).