Existence of perfect Mendelsohn designs with \(k=5\) and \(\lambda{}>1\)
Let \(v,k\), and \(\lambda\) be positive integers. A \((v,k,\lambda)\)- Mendelsohn design (briefly \((v,k,\lambda)-\text{MD})\) is a pair \((X,{\mathcal B})\) where \(X\) is a \(v\)-set (of points) and \({\mathcal B}\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks) such that every ordered pair of points of \(X\) is consecutive in exactly \(\lambda\) blocks of \({\mathcal B}\). A set of \(k\) distinct elements \(\{a_ 1,a_ 2,\dots,a_ k\}\) is said to be cyclically ordered by \(a_ 1<a_ 2<\cdots<a_ k<a_ 1\) and the pair \(a_ i\), \(a_{i+t}\) is said to be \(t\)-apart in cyclic \(k\)-tuple \((a_ 1,a_ 2,\dots,a_ k)\) where \(i+t\) is taken modulo \(k\). If for all \(t=1,2,\dots,k-1\), every ordered pair of points of \(X\) is \(t\)-apart in exactly \(\lambda\) blocks of \({\mathcal B}\), then the \((v,k,\lambda)\)-MD is called a perfect design and is denoted briefly by \((v,k,\lambda)\)-PMD. This paper is concerned mainly with the case where \(k=5\) and \(\lambda>1\). It is shown that the necessary condition for the existence of a \((v,5,\lambda)\)-PMD, namely, \(\lambda v(v-1)\equiv 0\pmod 5\), is also sufficient for \(\lambda>1\) with the possible exception of pairs \((v,\lambda)\) where \(\lambda=5\) and \(v=18\) or 28.
scientific article; zbMATH DE number 89919
scientific article