### Abstract

Let K ={K_{1},..., k_{m} } be a set of block sizes, and let {p_{1},..., p_{m}} be nonnegative numbers with Σ^{m}
_{ i=1}, p_{i} 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 k_{i}is p_{i},±ε for every i. We also show that the necessary conditions for a pairwise balanced design having precisely the fraction p_{i}, of its pairs in blocks of size k_{i} for each i are asymptotically sufficient.

Original language | English (US) |
---|---|

Pages (from-to) | 57-63 |

Number of pages | 7 |

Journal | Annals of Discrete Mathematics |

Volume | 42 |

Issue number | C |

DOIs | |

State | Published - 1989 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Annals of Discrete Mathematics*,

*42*(C), 57-63. https://doi.org/10.1016/S0167-5060(08)70097-9

**Percentages in Pairwise Balanced Designs.** / Colbourn, Charles; Rötdl, Vojtech.

Research output: Contribution to journal › Article

*Annals of Discrete Mathematics*, vol. 42, no. C, pp. 57-63. https://doi.org/10.1016/S0167-5060(08)70097-9

}

TY - JOUR

T1 - Percentages in Pairwise Balanced Designs

AU - Colbourn, Charles

AU - Rötdl, Vojtech

PY - 1989

Y1 - 1989

N2 - Let K ={K1,..., km } be a set of block sizes, and let {p1,..., pm} be nonnegative numbers with Σm i=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 Σm i=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.

UR - http://www.scopus.com/inward/record.url?scp=77957073560&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957073560&partnerID=8YFLogxK

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 -