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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:
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