Discrete Mathematics 67 (1987) 241-247 North-Holland 241 INTERSECTION PROBLEM OF STEINER SYSTEMS S(3, 4, 2v) Hung-Lin FU Department of Mathematics, Auburn University, Auburn, A L 36849, U.S.A. Received 3 March 1986 Revised 6 October 1986 1. Introduction A Steiner quadruple system of order v (SQS(v)) is a pair (Q, q), where Q is a v-set and q is a collection of 4-element subsets of Q, called blocks, such that every 3-element subset of Q is contained in exactly one block of q. Hanani [12] proved that an SQS(v) exists if and only if v =-2 or 4 (mod 6). It is easy to see that [ql = ~v(v - 1)(v - 2), which we will denote by q~ in what follows. Denote by J[v] the set of all positive integers k such that there exists a pair of SQS(v) which have exactly k blocks in common, and set I[v]= {0, 1, 2,..., q~, - 14} U {qv - 12, q~ - 8, qo}. In [6], Gionfriddo and Lindner conjectured that J[v] = l[v] for every v - 2 or 4 (mod 6) and v/> 8. Since the conjecture by Giofiiddo and Lindner, a considerable amount of work has been done in an attempt to prove that J[v] = I[v] [3-11]. But compared to the whole problem, we still have a lot of work to do. In this paper, we proved that J[2v] = l[2v] for certain admissible order v of Steiner quadruple systems, and l[2v]\ {q2~ -h: h = 17, 18, 19} ~_J[2v] for every v -= 2 or 4 (mod 6). Moreover, with the assumption of J[16] = I[16], J[20] = I[20], and J[28] = I[28], we are able to prove that J[2v] = I[2v], where v- 2 or 4 (mod 6) and v 1> 4. 2. The main theorems A partial quadruple system (PQS) is a pair (P, p), where P is a finite set and p is a collection of 4-subsets of P (called blocks) such that every 3-subset of P is contained in at most one block of p. Two partial quadruple systems (P, Pl) and (P, P2) are said to be mutually balanced if any given triple of distinct elements of P is contained in a block in pl if and only if it is contained in a block of P2. Two mutually balanced PQSs are disjoint if they have no block in common. (DMB PQSs). It is easy to see if (P, P0 and (P, P2) are two DMB PQSs, then 0012-365X/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)