Maximal k-Sum-Free Collections in an Abelian Group

Let be an Abelian group of order n , let be an integer, and be nonempty subsets of . The collection is called -sum-free (abbreviated - SFC ) if the equation has no solutions in the collection where , …, . The family of - SFC in will be denoted by . The collection is called maximal by capacity if it is maximal by the sum of , and maximal by inclusion if for any and the collection Suppose In this work, we study the problem of the maximal value of . In particular, the maximal value of for the cyclic group is determined. Upper and lower bounds for are obtained for the Abelian group The structure of the maximal k -sum-free collection by capacity (by inclusion) is described for an arbitrary cyclic group.