## Abstract

We consider the question of the existence of pairwise balanced designs on n + s points containing a block of cardinality s and all other blocks of cardinality at least 3. We denote such a design by PBDH(n + s:s). Building on earlier work of Rees, Mendelsohn, Jungnickel and Lenz, we show that necessary conditions for the existence of a PBDH(n + s:s) with n even are 1 ≤ s ≤ n- 1 and s n- 2 when n = 2 or 4 (mod 6), and 1 ≤ s ≤ n- 1 when n = 0 (mod 6). We show that these conditions are also sufficient except in the following instances, when such a design does not exist: (n, s) ε ((6,2),(8,2),(10,2)). In the case when n is odd a necessary condition is that 1 ≤ s ≤ n(n- 1)/(n + 3). This condition is proved to be sufficient for all n ≥ 37. For n ≤ 35 we show that a PBDH(n + s:s) does not exist when (n, s) ε ((5,2),(7,2),(7,4).(11,2)) and that there are at most 16 other possible exceptional pairs (n, s).

Original language | English |
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Title of host publication | Graphs, Matrices, and Designs |

Publisher | CRC Press |

Pages | 171-204 |

Number of pages | 34 |

ISBN (Electronic) | 0824787900, 9781351444385 |

ISBN (Print) | 9781138403987 |

DOIs | |

State | Published - 1 Jan 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 1993 by MARCEL DEKKER, INC. All Rights Reserved.

## ASJC Scopus subject areas

- General Mathematics