Igor Protasov, Sergii Slobodianiuk
November 29, 2014
Let G be a group and let κ be a cardinal. A subset A of G is called left (right) κ-large if there exists a subset F of G such that | F | < κ and G = F A ( G = A F ). We say that A is κ-large if A is left and right κ-large. It is known that every infinite group G can be partitioned into countably many ℵ 0 -large subsets. On the other hand, every amenable (in particular, Abelian) group G cannot be partitioned into more than ℵ 0 ℵ 0 -large subsets. We prove that every group G of cardinality κ can be partitioned into κ ℵ 1 -large subsets and every free group F κ in the infinite alphabet κ can be partitioned into κ 4-large subsets but cannot be partitioned into three 3-large subsets.