Maximal sum-free sets and block designs (Q2777539)

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.