  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.
