(Boca Raton, FL, U.S.A., 2018-02) Wilbroad, Bezire; Dinesh, G. Sarvate
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.
