Is there a group $G$ in which every abelian subgroup is finite and there is no upper bound on the sizes of its abelian subgroups?

Let me say that a counterpart of the question above has posed by Paul Erd\"os in 1975 as follows: Is there a group $G$ in which every subset consisting of pairwise non-commuting elements is finite and there is no upper bound on the sizes of such subsets?

The asnwer was given by B. H. Neumann in 1976 in negative. It was proved that groups $G$ satisfying the first assumption of the latter question must be center-by-finite and so the index of the center is a upper bound for sizes of subsets consisting of pairwise non-commuting elements.