# On infinite combinatorics of ultrafilters

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?

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).