In his famous paper "On a problem of Kurosh, Jonsson groups, and applications" of 1980, Shelah constructed a CH-example of an uncountable group $G$ equal to $A^{6641}$ for any uncountable subset $A\subset G$.

Let us call a group $G$

$\bullet$ *$n$-Shelah* if $G=A^n$ for each subset $A\subset G$ of cardinality $|A|=|G|$;

$\bullet$ *Shelah* if $G$ is $n$-Shelah for some $n\in\mathbb N$;

$\bullet$ *almost Shelah* if for each subset $A\subset G$ of cardinality $|A|=|G|$ there exists $n\in\mathbb N$ such that $A^n=G$;

$\bullet$ *Jonsson* if each subsemigroup $A\subset G$ of cardinality $|A|=|G|$ coincides with $G$.

$\bullet$ *Kurosh* if each subgroup $A\subset G$ of cardinality $|A|=|G|$ coincides with $G$.

It is clear that for any group $G$ the following implications hold:

finite $\Leftrightarrow$ 1-Shelah $\Rightarrow$ $n$-Shelah $\Rightarrow$ Shelah $\Rightarrow$ almost Shelah $\Rightarrow$ Jonsson $\Rightarrow$ Kurosh.

In the mentioned paper, Shelah constructed a ZFC-example of an uncountable Jonsson group and also a CH-example of an uncountable 6641-Shelah group.

Problem 1.Can an infinite (almost) Shelah group be constructed in ZFC?

Problem 2.Find the largest possible $n$ (which will be smaller than 6640) such that each $n$-Shelah group is finite.

This result of Protasov implies

Theorem (Protasov).Each countable Shelah group is finite.

It is easy to show that each 2-Shelah group is finite.

Problem 3.Is each 3-Shelah group finite?

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