### Abstract

Let K = {k_{1},...,k_{m}} be a set of block sizes, and let {p_{1},...,p_{m}} be nonnegative numbers with σ^{mi}=1p_{i} = 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 k_{i} is p_{i}±ε{lunate} 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) |
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Pages (from-to) | 57-63 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 77 |

Issue number | 1-3 |

DOIs | |

State | Published - Jan 1 1989 |

Externally published | Yes |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Colbourn, C. J., & Rodl, V. (1989). Percentages in pairwise balanced designs.

*Discrete Mathematics*,*77*(1-3), 57-63. https://doi.org/10.1016/0012-365X(89)90351-8