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 $A^{6643}$ 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 $n\in\mathbb N$;

$\bullet$ weakly 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 subgroup $A\subset G$ of cardinality $|A|=|G|$ coincides with $G$.

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

finite $\Rightarrow$ 1-Shelah $\Rightarrow$ $n$-Shelah $\Rightarrow$ Shelah $\Rightarrow$ weak Shelah $\Rightarrow$ Jonsson.

In the mentioned paper, Shelah constructed a ZFC-example of an uncountable Jonsson group and also a CH-example of an uncountable 6643-Shelah group.

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

Problem 2. Find an upper bound (better than 6643) for the smallest number $n$ for which there exists a (consistent) example of an $n$-Shelah group.