Finite embedding theorems for partial pairwise balanced designs

Abstract

The pair (P, p) is a (partial) (n, b)-PBD if (P, p) isa (partial) pairwise balanced design with the property that |P| = n and each block in p has exactly b elements. The following theorems are proved.

Theorem. If (P, p) is an (n, b)-PBD and n > b ⩾ 4, then (P, p) has an isomorphic disjoint mate. (Theorem 2.3)

Theorem. Suppose k and b are positive integers and b ⩾ 5. There is a constant C(k, b) such that if (P, p) is an (n, b)-PBD and n > C(k, b), then there exist k mutually disjoint isomorphic mates of (P, p). (Theorem 2.2)

Theorem. Suppose k and b are positive integers, k ⩾ 2 and b ⩾ 5. If (P, p₁, (P, p₂),…, (P, p_k) is a collection of partial (|P|, b)-PBD's, there exist k (n, b)-PBD's (X, x₁), (X, x₂), …, (X, x_k) such that (P, p₁) is embedded in (X, x₁) and for i ≠ j, p₁ ∩ p₁ = x₁ ∩ x₁. Additionally the existence of certain collections valuable in embedding is explored. (Theorem 4.10)