Аннотация:A collection (A_1,...,A_k) of subsets of a group G is called k-solution-free if the equation x_1 + ... + x_k = 0 has no solution in (A_1,...,A_k), where x_1 \in A_1,...,x_k \in A_k. The asymptotic behavior for the logarithm of the number of k-solution-free collections in Abelian groups is obtained.