Partitions of groups into large subsets

Let G be a group and let k be a cardinal. A subset A of G is called left (right) k-large if there exists a subset F of G such that |F| < { and G = FA (G = AF). We say that A is k-large if A is left and right k-large. It is known that every infinite group G can be partitioned into countably many \aleph_0-large subsets. On the other hand, every amenable (in particular Abelian) group G cannot be partitioned into > \aleph_0 \aleph_0-large subsets. We prove that every infinite group G of cardinality k can be partitioned into k left- \aleph_1-large subsets and every free group F_k in the infinite alphabet k can be partitioned into k 4-large subsets.