Elsevier

Discrete Mathematics

Volume 77, Issues 1–3, September–October 1989, Pages 57-63
Discrete Mathematics

Percentages in pairwise balanced designs

To Professor Haim Hanani on the occasion of his seventy-fifth birthday.
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Abstract

Let K = {k1,…,km} be a set of block sizes, and let {p1,…,pm} be nonnegative numbers with σmi=1pi = 1. We prove the following theorem: for any ϵ > 0, if a (v,K,1) pairwise balanced design exists and v is sufficiently large, then a (v,K,1) pairwise balanced design exists in which the fraction of pairs appearing in blocks of size ki is pi±ϵ for every i. We also show that the necessary conditions for a pairwise balanced design having precisely the fraction pi of its pairs in blocks of size ki for each i are asymptotically sufficient.

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