On the Existence of $(v,n,4,\lambda )$-$IPMD$ for Even $\lambda$

Abstract

Let $v$, $k$,$\lambda$ and $n$ be positive integers. $(x_1,x_2,\dots ,x_k)$ is defined to be $\{(x_1,x_2),(x_2,x_3),\dots ,(x_k-1,x_k),(x_k,x_1)\}$, and is called a cyclically ordered $k$-subset of $\{x_1,x_2,\dots ,x_1\}$. An incomplete perfect Mendelsohn design, denoted by $(v,n,4,\lambda )$-IPMD, is a triple $(X,Y,\mathcal{B})$, where $X$ is a $v$-set (of points), $Y$ is an $n$-subset of $X$, and $\mathcal{B}$ is a collection of cyclically ordered $k$-subsets of $X$ (called blocks) such that every ordered pair $(a,b)\in X\times X\setminus Y\times Y$ appears $t$-apart in exactly $\lambda$ blocks of $\mathcal{B}$ and no ordered pair $(x,y)\in Y\times Y$ appears in any block of $\mathcal{B}$ for any $t$, where $1\le t\le (k–1)$. In this paper, the necessary condition for the existence of a $(v,n,4,\lambda )$-IPMD for even $\lambda$, namely $v\ge (3n+1)$, is shown to be sufficient.