TY - JOUR

T1 - Percentages in pairwise balanced designs

AU - Colbourn, Charles J.

AU - Rodl, Vojtech

N1 - Funding Information:
Thanks to Alex Rosa for suggesting this problem, and to Kevin Phelps for helpful discussions. Research of the first author is supported by NSERC Canada under grant A0579.

PY - 1989

Y1 - 1989

N2 - 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{lunate} > 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{lunate} 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.

AB - 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{lunate} > 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{lunate} 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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U2 - 10.1016/0012-365X(89)90351-8

DO - 10.1016/0012-365X(89)90351-8

M3 - Article

AN - SCOPUS:50849147643

VL - 77

SP - 57

EP - 63

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -