Perfect Mendelsohn Designs With Block Size Four

Let \(v\), \(k\), and \(\lambda\) be positive integers. A \((v, k, \lambda)\)-Mendelsohn design (briefly \((v,k,\lambda)\)-MD) is a pair \((X,B)\) where \(X\) is a \(v\)-set (of points) and \(B\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks) such that every ordered pair of points of \(X\) are consecutive in exactly \(\lambda\) blocks of \(B\). A set of $\delta$ distinct elements \(\{a_1,a_2,…,a_\delta\}\) is said to be cyclically ordered by \(a_1<a_2<…<a_k<a_1\) and the pair \(a_i,a_{i+t}\) are said to be \(t\)-apart in a cyclic \(k\)-tuple \((a_1,a_2,…,a_k)\) where \(i+ t\) is taken modulo \(k\). If for all \(t=1,2,…, k-1\), every ordered pair of points of \(X\) are \(t\)-apart in exactly \(\lambda\) blocks of \(B\), then the \((v,k,\lambda)\)-MD is called a perfect design and is denoted briefly by \((v,k,\lambda)\)-PMD. A necessary condition for the existence of a \((v,k,\lambda)\)-PMD is \(\lambda v(v-1)\equiv0\) (mod \(k\)). In this paper, we shall be concerned mainly with the case where \(k=4\). It will be shown that the necessary condition for the existence of a \((v,4,\lambda)\)-PMD, namely, \(\lambda v(v-1)\equiv0\) (mod \(4\)), is also sufficient, except for \(v=4\) and \(\lambda\) odd, \(v=8\) and \(\lambda=1\), and possibly excepting \(v=12\) and \(\lambda=1\). Apart from the existence of a \((12,4,1)\)-PMD, which remains very much in doubt, the problem of existence of \((v,4,\lambda)\)-PMDs is now completely settled.