Simple neighbourhoods in triple systems (Q749540)

For a fixed element x in a triple system with element set V, the neighborhood N(x) of x is the multigraph with vertex set V-\(\{\) \(x\}\) and an edge joining y and z whenever y and z are in a triple with x for all triples in the system. The author proves that for every \(\lambda\), every \(\lambda\)-regular graph meeting certain obvious necessary conditions occurs as a neighborhood in some triple system. He also makes progress on the conjecture that every \(\lambda\)-regular multigraph meeting the same necessary conditions is a neighborhood.