Multi-Set designs and numbers of common triples

A two-fold multi-set triple system is a collection of triples chosen, possibly with repetitions, from av-set in such a way that each pair (whether distinct or not) occurs precisely two times. For instance the triplex, x, y (which we shall write asxxy) contains the pairxx once and the pairxy twice. Such a design has v(v + l)/3 blocks, and a necessary and sufficient condition for existence is thatv = 0 or 2 (mod 3) and y≥5. The design is called simple if it contains no repeated blocks. Let I₂(v) denote the set of non-negative integerss such that there exist two simple two-fold multi-set triple systems of orderv with preciselys blocks in common. In this paper we determine I₂(v) for all admissiblev.