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 five years earlier, in Prague.
Submitted to Forum of Mathematics, Pi, July 2023.
Accepted, February 2025.