Let $\mathsf C_n$ denotes the statement:

*for any family $\mathcal F$ of $n$-element sets there exists a choice function (i.e., a function $f:\mathcal F\to\bigcup\mathcal F$ such that $f(F)\in F$ for all $F\in\mathcal F$).*

It is known that $\mathsf C_2\Rightarrow \mathsf C_4$ in ZF.

This fact suggests introducing a partial preorder $\preceq$ on the set $\mathbb N$ of positive integers defined by $n\preceq m$ if $\mathsf C_m\Rightarrow \mathsf C_n$ in ZF.

Also we can write that $n\cong m$ if $\mathsf C_n\Leftrightarrow \mathsf C_m$.

It is easy to show that $n\preceq m$ if $n$ divides $m$. So, $1\preceq n$ for any $n\in\mathbb N$ and $2\preceq n$ for any even number $n$.

On the other hand, $\mathsf C_2\Rightarrow \mathsf C_4$ implies that $2\cong 4$.

What else is known about the partial preorder $\preceq$? Maybe there exists a precise (arithmetic) description of this preorder.

A more specific question: is $2^n\cong 2$ for any $n\in\mathbb N$?

I know that similar questions were studied by Mostowski, Tarski, Truss, Jech so maybe the answer is already known?

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