## Abstract

When a v-set can be equipped with a set of k-subsets so that every 2-subset of the v-set appears in exactly (or at most, or at least) one of the chosen k-subsets, the result is a balanced incomplete block design (or packing, or covering, respectively). For each k, balanced incomplete block designs are known to exist for all sufficiently large values of v that meet certain divisibility conditions. When these conditions are not met, one can ask for the packing with the most blocks and/or the covering with the fewest blocks. Elementary necessary conditions furnish an upper bound on the number of blocks in a packing and a lower bound on the number of blocks in a covering. In this paper it is shown that for all sufficiently large values of v, a packing and a covering on v elements exist whose numbers of blocks differ from the basic bounds by no more than an additive constant depending only on k.

Original language | English (US) |
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Pages (from-to) | 1440-1449 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 120 |

Issue number | 7 |

DOIs | |

State | Published - Sep 2013 |

## Keywords

- Balanced incomplete block design
- Group divisible design
- Pair covering
- Pair packing
- Pairwise balanced design

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics