Solving a conjecture of Sedlacek: Maximal edge sets in the 3-uniform sumset hypergraphs (Q1322279)

For integers \(n\geq 3\), the hypergraph \(S_ n\) has \(\{1,2,\dots,n\}\) as vertex set, and all sets of the form \(\{i,j,i+ j\}\) with \(1\leq i,j,i+ j\leq n\) as edges. Surely \(\bigl\lfloor{n\over 3}\bigr\rfloor\) is an upper bound for the maximum number \(\nu(S_ n)\) of pairwise disjoint edges in \(S_ n\). Since the sum of the numbers in every edge is even, this bound can be improved to \(\bigl\lfloor{n\over 3}\bigr\rfloor- 1\) for \(n\equiv 6\) or \(n\equiv 9\) modulo 12. It is shown that these bounds are indeed the exact values of \(\nu(S_ n)\).