A Shelah group in ZFC

Source abstract

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 Jónsson group , and the latter is called a Shelah group . In this paper, we construct a Shelah group on the grounds of $\textsf {{ZFC}}$ 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 we 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.