This is a generalisation of an older [question inspired by a football tournament](https://mathoverflow.net/questions/432988/on-a-combinatorial-design-inspired-by-a-football-soccer-tournament) (which does not have an answer yet).

Let $\frak P$ be a partition of $\omega$ into blocks, that is, pairwise disjoint sets, such that every block has at least $2$ elements. (Blocks may be infinite.) Assume that $\frak P$ consists of infinite many blocks.

Is there for every $n\in\mathbb{N}$ a bijection $\varphi_n:\omega\to\omega$, such that the following condition is met?
> Whenever $a\neq b\in\omega$ there is a unique $n\in\mathbb{N}$ such that $\{\varphi_n(a), \varphi_n(b)\}\subseteq B$ for some $B\in{\frak P}$.