An Existence Theory for Incomplete Designs

Canadian mathematical bulletin

Alan C.H. Ling³  Peter Dukes¹  E.R. Lamken² 

[1] Department of Mathematics and Statistics, University of Victoria, Victoria, Canada;773 Colby Street, San Francisco, CA, USA 94134;Department of Computer Science, University of Vermont, Burlington, VT, USA 05405

关键词: block design;    hypergraph;   

DOI  :  10.4153/CMB-2015-073-7

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

An incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a `hole'. If there are $v$ points, a hole of size $w$, and all (other) block sizes equal $k$, this is denoted IPBD$((v;w),k)$. In addition to congruence restrictions on $v$ and $w$, there is also a necessary inequality: $v gt (k-1)w$. This article establishes two main existence results for IPBD$((v;w),k)$: one in which $w$ is fixed and $v$ is large, and the other in the case $v gt (k-1+epsilon) w$ when $w$ is large (depending on $epsilon$). Several possible generalizationsof the problem are also discussed.

【 授权许可】

Unknown