A Shelah group in ZFC

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 $\kappa ={\mathrm{\aleph }}_1,{\mathrm{\aleph }}_2,{\mathrm{\aleph }}_3,\dots$).
This also yields the first consistent example of a Shelah group of size a limit cardinal.

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