Cycling through a general combinatorial design on $\omega$ 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.) 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 $B\in{\frak P}$.

 A: If there are infinitely many blocks, then indeed you can find the permuations $\pi_n$ as desired. Indeed, there are computable such $\pi_n$. Furthermore, you don't need that every block has size at least two, but rather only that there is at least one block of size at least two.
We build the sequence of permuations $\pi_n$ in stages, so that at each stage, only finitely much of finitely many permutations is specified. We want gradually to fulfill the following requirements:

*

*for each $n$ and $k$, the value of $\pi_n(k)$ is specified

*for each $n$ and $k'$, there is some $k$ such that $\pi_n(k)=k'$

*for each pair $a\neq b$, there is a unique $n$ such that $\pi_n(a)$ and $\pi_n(b)$ appear in the same block.

Now, the point is that we can gradually fulfill these requirements in a computable process that specifies more and more of the system of permuations. We can always fulfill the first requirement, for example, without any tension with the other requirements by specifying $\pi_n(k)$ to have a value appearing in a totally new block not yet used. We can fulfill the second requirement by using a value $k$ not yet mentioned in the construction. Infinitely often, we can give attention to the third requirement by selecting for a given pair $a\neq b$ a totally new $n$ not yet considered at all and placing $\pi_n(a)$ and $\pi_n(b)$ to have values in the same block. At each later stage, we can preserve this uniqueness property for this pair.
In this way, the permutations are gradually defined, while fulfilling the properties, and so there are indeed such permuations.
