Super-magic complete \(k\)-partite hypergraphs (Q5936101)

A complete \(k\)-partite hypergraph \({\mathbf H}^k_n\) is a hypergraph whose vertex set is partitioned into \(k\) pairwise disjoint \(n\)-element subsets called independent sets and the edge set consists of all \(k\)-element sets of vertices intersecting each of the independent sets (on exactly one element). Clearly \({\mathbf H}^k_n\) has \(n^k\) edges. A hypergraph \({\mathbf H}^k_n\) is called super-magic if one can label its edges with the consecutive integers \(1,2,\dots, n^k\) such that for any choice of \(k-1\) particular vertices, each from a different independent set, the sum of labels of all \(n\) edges containing the \(k-1\) vertices is the same. The author shows that for all positive integers \(n\neq 2,6\) and \(k\geq 2\), the complete \(k\)-partite hypergraph \({\mathbf H}^k_n\) is super-magic.