Bipacking pairs by quintuples: The case \(\nu\equiv 13\bmod 20\)
A \((\nu,\kappa,\lambda)\) packing design of order \(\nu\), block size \(\kappa\), and index \(\lambda\) is a collection of \(\kappa\)-element subsets, called blocks, of a \(\nu\)-set \(V\) such that every 2-subset of \(V\) occurs in at most \(\lambda\) blocks. The packing problem is to determine the maximum number of blocks in a packing design. Packing with \(\lambda= 2\) is called bipacking. In this paper, the author solved the bipacking problem in the case \(\kappa= 5\) and \(\nu\equiv 13\pmod {20}\). Specifically he proved the following main Theorem: For all \(\nu\equiv 13\pmod {20}\), \(\nu\neq 13\), we have \(\sigma(\nu)= \psi(\nu)\), and \(\sigma(13)= \psi(13)- 1\), where \(\sigma(\nu)= \sigma(\nu,5,2)\) denotes the maximum number of blocks in a \((\nu,\kappa,\lambda)\) packing design, \(\psi(\nu,\kappa,\lambda)=\left[{\nu\over \kappa}\left[{\nu-1\over \kappa- 1} \lambda\right]\right]\), \(\psi(\nu)= \psi(\nu,5,2)\), and \([x]\) is the floor of \(x\).
scientific article; zbMATH DE number 681687