On cc-Bhaskar Rao Designs and tight embeddings for path designs

Under the right conditions it is possible for the ordered blocks of a path design PATH(v,k,μ)PATH(v,k,μ) to be considered as unordered blocks and thereby create a BIBD(v,k,λ)BIBD(v,k,λ). We call this a tight embedding. We show here that, for any triple system TS(v,3)TS(v,3), there is always such an embedding and that the problem is equivalent to the existence of a (-1)-BRD(v,3,3)(-1)-BRD(v,3,3), i.e., a cc-Bhaskar Rao Design. That is, we also prove the incidence matrix of any triple system TS(v,3)TS(v,3) can always be signed to create a (-1)-BRD(v,3,3)(-1)-BRD(v,3,3) and, moreover, the signing determines a natural partition of the blocks of the triple system making it a nested design.