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 ∣Fg∩A∣⩽m for every g∈G∖K. We
show that each m-thin subset of a group G of cardinality ℵn, n=0,1,... can be partitioned into ⩽mn+1 1-thin subsets. On the
other side, we construct a group G of cardinality ℵω and point
out a 2-thin subset of G which cannot be finitely partitioned into 1-thin
subsets