The embedding of partial triple systems when 4 divides lambda

We show that if 4 divides I, then any partial triple system of order r and index 1 can be embedded in a proper triple system of index I and order n whenever n is I-admissible and n > 2r + 1. Moreover we find a set of necessary conditions for the embedding of a partial triple system of index I when I is even and show that when 4 divides 1, then a very closely related set of conditions is sufficient. A partial triple system of order r and index 2, a PTS(r, I) for short, is a collection of triples of elements of an r-set such that each pair of elements is in at most 2 of the triples. Such a PTS(r, A) is maximal if no further triples can be added to the collection without contravening one of the rules. If each pair of elements is in exactly ,? triples, then we have a triple system of order r and index I, a TS(r, A) for short. It is well known that these exist if and only if r 2 3 and the following two conditions are satisfied (see [ 161, for example): Ar(r-l)=O (mod 3)