Maximal k-Sum-Free Collections in an Abelian Groupстатья
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Аннотация:Let G be an Abelian group of order n, let k≥2 be an integer, and A_1,…,A_k be non-empty subsets of G. The collection (A_1,…,A_k ) is called k-sum-free (abbreviated k-SFC) if the equation x_1+···+x_k=0has no solutions in the collection (A_1,…,A_k ), wherex_1∈A_1,...,x_k∈A_k. The family of k-SFC in G will be denoted by SFC_k (G). The collection (A_1,…,A_k )∈SFC_k (G) is called maximal by capacity if it is maximal by the sum of |A_1 |+···+|A_k |, and maximal by inclusion if for any i∈{1,...,k} and x∈G \ A_i, the collection (A_1,...,A_(i-1),A_i∪{x},A_(i+1),...,A_k )∉SFC_k (G). Suppose ϱ_k (G)=|A_1 |+···+|A_k |. In this work, we study the problem of the maximal value of ϱ_k (G). In particular, the maximal value of ϱ_k (Z_d ) for the cyclic group Z_d is determined. Upper and lower bounds for ϱ_k (G) are obtained for the Abelian group G. The structure of the maximal k-sum-free collection by capacity (by inclusion) is described for an arbitrary cyclic group.