This is a generalisation of an older question inspired by a football 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.)

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 $B\in{\frak P}$.