Finite embedding theorems for partial pairwise balanced designs |
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Authors: | William B Poucher |
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Institution: | Dept. of Mathematics, David Lipscomb College, Nashville, Tennessee 37203, U.S.A. |
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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, pk) is a collection of partial (|P|, b)-PBD's, there exist k (n, b)-PBD's (X, x1), (X, x2), …, (X, xk) 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) |
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