Subdesigns in Steiner quadruple systems (original) (raw)

1991, Journal of Combinatorial Theory, Series A

Steiner quadruple system of order v, denoted SQS(v), is a pair (X, a), where X is a set of cardinality v, and 1 is a set of 4-subsets of C (called blocks), with the property that any 3-subset of X is contained in a unique block. If (X, 0) is an SQS(v) and (Y, C) is an SQS(w) with YE X and Cc@, we say that (Y, C) is a subdesign of (X, 0). Hanani has shown that an SQS(v) exists for all v E 2 or 4 (mod 6) and when v E {0, 1); such integers v are said to be admissible. A necessary condition for the existence of an SQS(v) with a subdesign of order w is that v = w or v > 2w. In this paper we show the existence of an explicitly computable constant k (independent of w) such that for all admissible v and all admissible w with v > kw there exists an SQS(v) containing a subdesign of order w. We also show that for any sufficiently large w we can take k = 12.54. To establish these results we introduce several new constructions for SQS, and we also consider the subdesign problem for related classes of designs. 0 1991 Academic press, h. 1. INTR~D~JcTI~N This paper is concerned with the existence problem for SQS(v) with a subdesignoforder w. LetA,=(O,l}u{u~2:~~2or4(mod6)) denote the set of admissible integers. For w E A,, we define S, to be the set of all orders u for which there exists an SQS(u) with a subdesign of order w. Define s, to be the least admissible integer vO, such that for all u EAT, u 2 u0 we have u E S,, if such a u0 exists; otherwise s, = co. In Section 3 we shall show that s, is finite for all w E A,, and then in Section 4 we shall show that s, < kw for some absolute constant k. In Section 5 we improve the constant k. To do this we need constructions for SQS(u) with subdesigns. These constructions are described in Section 2, including a review of existing constructions as applied to the subsystem problem. Several new constructions are also described in Section 2, and the details of these new results are given in Section 6. In Section 7 we give some generalizations of our results and, in Section 8, pose some problems.