TY - JOUR

T1 - Percentages in Pairwise Balanced Designs

AU - Colbourn, Charles J.

AU - Rötdl, 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 - 1989/1/1

Y1 - 1989/1/1

N2 - Let K ={K1,…, km} be a set of block sizes, and let {p1,…, pm} be nonnegative numbers with Σmi=1, pi 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 kiis 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.

AB - Let K ={K1,…, km} be a set of block sizes, and let {p1,…, pm} be nonnegative numbers with Σmi=1, pi 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 kiis 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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U2 - 10.1016/S0167-5060(08)70097-9

DO - 10.1016/S0167-5060(08)70097-9

M3 - Article

AN - SCOPUS:77957073560

VL - 42

SP - 57

EP - 63

JO - Annals of Discrete Mathematics

JF - Annals of Discrete Mathematics

SN - 0167-5060

IS - C

ER -