### Abstract

The minimum number of blocks having maximum size precisely four that are required to cover, exactly ? times, all pairs of elements from a set of cardinality v is denoted by g? (4)(v) (or g(4)(v) when ? = 1). All values of g? (4) (v) are known except for ? = 1 and v = 17 or 18. It is known that 30)? g(4)(l7) ? 31 and 32 ? g(4)(l8) ? 33. In this paper we show that g(4)(17?30 and g(4)(18) ? 32, thus finalising the determination of g? (4)v) for all ? and v.

Original language | English |
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Pages (from-to) | 303-313 |

Number of pages | 11 |

Journal | Australasian Journal of Combinatorics |

Volume | 36 |

Publication status | Published - 2006 |

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## Cite this

Gruettmueller, M., Roberts, I., D'arcy, S., & Egan, J. (2006). The minimum number of blocks in pairwise balanced designs with maximum block size 4: the final cases.

*Australasian Journal of Combinatorics*,*36*, 303-313.