The intersection problem for minimum coverings of Kn by triples

A Steiner triple system (more simply, triple system) is a pair T) where S is the vertex set of the complete undirected graph Kn on n vertices and T is a collection of edge-disjoint triangles (triples) which partition the edge set of Kn. The number n is called the order of the triple system (S, T) and it has been known forever since 1847 of triple systems (= set of all n such that a triple system of order n set of all n == 1 or 3 (mod 6). In this case ITI = n(n 1)/6. that the spectrum is precisely the In [3J C. C. Lindner and A. Rosa gave a complete solution of the intersection problem for triple systems by determining all pairs (n, k) such that there exists a pair of triple systems (S,T l) and (S, T 2) of order n such that ITl nT21 = k. In particular, if we set l[nJ {kl there exist a pair. of triple systems of order n intersecting in k triples }, thenA maximum packing of Kn with triples (M PT) is a pair (S, P), where S is the vertex set of Kn and P is a collection of edge-disjoint triangles (or triples) of the edge set of Kn so that is a large as possible. As with triple systems, the number n is called the order. The collection of unused edges is called the leave of the M PT (S, P). So, a triple system is aM PT with leave L 0. Just as with triple systems, nonisomorphic M PTs are like grains of sand on the beach. However, M PTs of the same order all have one thing in common; the leave! In particular if P) is aM PT of order n, then the leave is (i) a I-factor if n 0 or