Existence of rr-fold perfect (v,K,1)(v,K,1)-Mendelsohn designs with K⊆{4,5,6,7}K⊆{4,5,6,7}

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