Existence of -fold perfect -Mendelsohn designs with

Let v be a positive integer and let K be a set of positive integers. A ( v , K , 1 ) -Mendelsohn design, which we denote briefly by ( v , K , 1 ) -MD, is a pair ( X , B ) where X is a v -set (of points) and B is a collection of cyclically ordered subsets of X (called blocks) with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B . If for all t = 1 , 2 , … , r , every ordered pair of points of X are t -apart in exactly one block of B , then the ( v , K , 1 ) -MD is called an r -fold perfect design and denoted briefly by an r -fold perfect ( v , K , 1 ) -MD. If K = { k } and r = k − 1 , then an r -fold perfect ( v , { k } , 1 ) -MD is essentially the more familiar ( v , k , 1 ) -perfect Mendelsohn design, which is briefly denoted by ( v , k , 1 ) -PMD. In this paper, we investigate the existence of r -fold perfect ( v , K , 1 ) -Mendelsohn designs for a specified set K which is a subset of {4, 5, 6, 7} containing precisely two elements.