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Maximal sum-free sets and block designs. (English) Zbl 0991.20038
Let \(V\) be a finite set. A system \(\{B_1,\dots,B_b\}\) of subsets of \(V\) is called a block design if there exist positive integers \(k\) and \(r\) such that each \(B_i\) has \(k\) elements and each \(x\in V\) lies in \(r\) of the subsets \(B_1,\dots,B_b\). A nonempty subset \(S\) of a finite additive group \(G\) is said to be sum-free if \((S+S)\cap S=\emptyset\). The maximal sum-free subsets of \(G\) are defined in the natural way. The author proves the following result: If \(G\) is the cyclic group \(G_{p^n}\), where \(p\) is an odd prime congruent to 2 modulo 3 and \(n\geqq 1\), then the maximal sum-free sets of \(G\) form a block design.
MSC:
20K01 Finite abelian groups
05B05 Combinatorial aspects of block designs
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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References:
[1] CHIN A. Y. M.: On maximal sum-free sets of certain cyclic groups and block designs.
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