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Let ${\frak P}\subseteq {\cal P}(\omega)$ be a partition such that every block $B\in {\frak P}$ contains at least two integers.

Is there a countable set ${\cal F}$ of bijections $\varphi:\omega\to\omega$ such that given any two integers $a\neq b\in\omega$ there is exactly one $\varphi_{a,b}\in {\cal F}$ such that $\varphi_{a,b}(a)$ and $\varphi_{a,b}(b)$ belong to the same block $B\in{\frak P}$?

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    $\begingroup$ say, if there are only two blocks: $\{1,2\}$ and (everything else), one bijection is not enough, but for two bijections there exist two (and even infinitely many) elements which are mapped to the huge block in both. $\endgroup$
    – Fedor Petrov
    Commented Jan 7, 2023 at 21:21
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    $\begingroup$ Isn't this question the same as your question last autumn? mathoverflow.net/q/433364/1946 I answered there affirmatively if there are infinitely many blocks. $\endgroup$
    – Joel David Hamkins
    Commented Jan 8, 2023 at 12:13

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