13.44 (1995)

For any partition of an arbitrary group $G$ into finitely many subsets $G = A_1 \cup \dots \cup A_n$, there exists a subset of the partition $A_i$ and a finite subset $F \subseteq G$, such that $G = A_i^{-1}A_i F$ (I. V. Protasov, Siberian Math. J., 34, no. 5 (1993), 938–952). Can the subset $F$ always be chosen so that $|F| \leqslant n$? This is true for amenable groups.