Geodesic
A Shelah group in ZFC
Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e13

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Poór, Márk; Rinot, Assaf. A Shelah group in ZFC. Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e13. doi: 10.1017/fmp.2025.5
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