Partial Triple Systems With Equal-Sized Holes

Let λ be a positive integer. A group divisible design(GDD) of index λ is a triple(X, G, B), where X is a finite set of points, G is a partition of X into subsets called groups, B is a collection of subsets of X(called blocks), such that a group and a block contain at most one common point, and every pair of points from distinct groups occurs in exctaly λ blocks. GDDs are a kind of usefull designs in combinatorial design theory, which plays an important role in the construction of the other kinds of designs. The main purpose of this paper is to determine the existence of(3, λ)-GDDs of type ur1 t. When λ = 1, the existence has been setted by Colbourn et al, but we find that there are some errors in their proof. In this paper we propose a new construction for(3, λ)-GDDs of type ur1 tand fix the problems by applying the new construction, we also solve the existence problem for the GDDs with the case of λ > 1. The following is the main result of this paper.Let u, r, t and λ be positive integers. The necessary and sufficient conditions for the existence of a(3, λ)-GDD of type ur1 tare that(i) λ(u- 1) ≡ 0(mod 2);(ii) λ(r + t- 1) ≡ 0(mod 2);(iii) if r = 1, then t ≥ u + 1;(iv) if r = 2, then t ≥ u;(v) λt2+ rut +r2u2≡ 0(mod 3).expect for(u, r, t, λ) =(2, 2, 2, 0(mod 6)).