Pairwise balanced designs with holes

Alan Hartman, Katherine Heinrich

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


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 languageEnglish
Title of host publicationGraphs, Matrices, and Designs
PublisherCRC Press
Number of pages34
ISBN (Electronic)0824787900, 9781351444385
ISBN (Print)9781138403987
StatePublished - 1 Jan 2017
Externally publishedYes

Bibliographical note

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

ASJC Scopus subject areas

  • General Mathematics


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