Thin Subsets of Groups - Protasov, I. V.; Slobodianiuk, S. V. (Ukrains’kyi Matematychnyi Zhurnal, 2013)
For a group G and a natural number m; a subset A of G is called m-thin if, for each finite subset F of G; there exists a finite subset K of G such that |F g ∩ A| ≤ m for all g ∈ G K: We show that each m-thin subset of an Abelian group G of cardinality ℵ n ; n = 0, 1,… can be split into ≤ m n+1 1-thin subsets. On the other hand, we construct a group G of cardinality ℵ ω and select a 2-thin subset of G which cannot be split into finitely many 1-thin subsets.