Mandatory representation designs (Q1112825)

A pairwise balanced design (PBD) is made up of a set of points X and a collection of subsets B of X such that any pair of distinct points occurs in a unique block. A PBD (K;\(\nu)\) is a pairwise balanced design on \(\nu\) points such that the size of each block is an element of K. A mandatory representation design MR (K;\(\nu)\) is a PBD (K;\(\nu)\) such that for each k in K there is a block in the design whose size is k. In this paper the authors consider the existence of MR (K;\(\nu)\)'s when \(3\in K\). A complete spectrum for MR (3,k;\(\nu)\), k an odd integer, is determined. It is pointed out that MR (3,k;\(\nu)\), k being even, is still an open problem.