Joint work with Márk Poór.
Abstract. In a paper from 1980, Shelah constructed an uncountable group all of whose proper subgroups are countable. Assuming the continuum hypothesis, he constructed an uncountable group $G$ that moreover admits an integer $n$ satisfying that for every uncountable $X\subseteq G$, every element of $G$ may be written as a group word of length $n$ in the elements of $X$. The former is called a Jonsson group and the latter is called a Shelah group.
In this paper, we construct a Shelah group on the grounds of ZFC alone, that is,
without assuming the continuum hypothesis. More generally, we identify a combinatorial condition (coming from the theories of negative square-bracket partition relations and strongly unbounded subadditive maps) sufficient for the construction of a Shelah group of size $\kappa$, and prove that the condition holds true for all successors of regular cardinals (such as $\kappa=\aleph_1,\aleph_2,\aleph_3,\ldots$).
This also yields the first consistent example of a Shelah group of size a limit cardinal.
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Theorem A answers affirmatively a problem posed at https://mathoverflow.net/q/313516/61536
Thank you, Taras. We will add a mention to this, though the problem is considerably older. I had an email exchange about the problem of constructing a “boundedly Jonsson” group in ZFC with Ol’ga Sipacheva back in May 2006, and she mentioned discussing it with Shelah fiver years earlier, in Prague.