In his famous paper ["On a problem of Kurosh, Jonsson groups, and applications"](https://www.sciencedirect.com/science/article/pii/S0049237X08713466) of 1980, Shelah constructed a CH-example of an uncountable group $G$ equal to $A^{6640}$ 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:



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



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> **Problem 1.** Can an infinite (almost) Shelah group be constructed in ZFC?
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> **Problem 2.** Find the largest possible $n$ (which will be smaller than [6640](https://mathoverflow.net/questions/314007/on-the-number-n-0-in-shelahs-construction-of-a-jonsson-group)) such that each $n$-Shelah group is finite.
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[This result](https://link.springer.com/article/10.1007/s00012-010-0032-0) of Protasov implies



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> **Theorem (Protasov).** Each countable Shelah group is finite.
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It is easy to show that each 2-Shelah group is finite.



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> **Problem 3.** Is each 3-Shelah group finite?
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A Shelah group in ZFC?
In [this preprint](https://arxiv.org/pdf/2305.11155.pdf), Mark Poor and Assaf Rinot construct a ZFC-example of a $10120$-Shelah group $G$ of cardinality $|G|=\lambda^+$, where $\lambda$ is an arbitrary regular сardinal.


