2023/05/18 by Márk Poór, Assaf Rinot, Poór, Márk +1 · 1 citation
Mathematics · #20A15 #20E15 #20F06 #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Primary 03E75 #Rings, Modules, and Algebras #Secondary 03E02
paper · pdf · doi:10.48550/arxiv.2305.11155
openalex publication_date 2023/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 κ, and prove that the condition holds true for all successors of regular cardinals (such as κ=ℵ1,ℵ2,ℵ3,…). This also yields the first consistent example of a Shelah group of size a limit cardinal.