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}$$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:
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?
