A Shelah group in ZFC

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 ⊆ 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 Jónsson 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 K, and we prove that the condition holds true for all successors of regular cardinals (such as K = ℵ₁, ℵ₂, ℵ₃, ...). This also yields the first consistent example of a Shelah group of size a limit cardinal.