On The Number of Partial Steiner Systems

We give a simple proof of the result of Grable on the asymptotics of the number of partial Steiner systems S(t,k,m). # 2000 John Wiley & Sons, Inc.J Combin Designs 8:347±352, 2000 Keywords: partical Steiner system; matching; hypergraph 1. INTRODUCTION A partial Steiner system S…t; k;m† is a collection of k-subsets of an m-element set M such that each t-subset is contained in at most one k-subset from S…t; k;m†. When every t-subset of M is contained in exactly one k-subset from S…t; k;m†, we have a classical Steiner system on the set M with parameters t and k. Some bounds of the number of such systems for t ˆ 2; k ˆ 3 and t ˆ 3; k ˆ 4 were obtained in [1], [9], [7] and [6]. Very little is known about the number of classical Steiner systems for large t and k. The number of distinct partial Steiner systems S…t; k;m† we denote by s…t; k;m†. For two sequences fm and gm we write fm gm if fm=gm ! 1 as m!1. In [5] Grable announced that using the RoÈdl nibble algorithm [8] and generalizing the result in [3] he proved the following: Theorem 1. Let t and k be two ®xed positive integers, t < k. Then ln s…t; k;m† k ÿ t …k†t m ln m as m!1; where …k†t ˆ k…k ÿ 1† . . . …k ÿ t ‡ 1†.