BULLETIN of the Volume 82 February 2018 INSTITUTE of COMBINATORICS and its APPLICATIONS

Abstract

We define a 3-GDD(n, 2, k, λ1, λ2) by extending the definitions of a group divisible design and a t-design and give some necessary conditions for its existence. We prove that these necessary conditions are sufficient for the existence of a 3-GDD(n, 2, 4, λ1, λ2) except possibly when n ≡ 1, 3 (mod 6), n 6= 3, 7, 13 and λ1 > λ2. It is known that a partition of all 3-subsets of a 7-set into 5 Steiner triple systems (a large set for 7) does not exist, but we show that the collection of all 3-sets of a 7-set along with a Steiner triple system on the 7-set can be partitioned into 6 Steiner triple systems. Such a partition is then used to prove the existence of all possible 3-GDDs for n = 7.