Existence of perfect Mendelsohn designs with k=5 and λ>1

AbstractLet υ, k, and λ be positive integers. A (υ, k, λ)-Mendelsohn design (briefly (υ, k, λ)-MD) is a pair (X, B) where X is a υ-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 λ blocks of B. A set of k distinct elements {a1, a2,…, ak} is said to be cyclically ordered by a1<a2<⋯<ak<a1 and the pair ai, ai+t is said to be t-apart in cyclic k-tuple (a1, a2,…, ak) where i+t is taken modulo k. It for all t=1,2,…, k-1, every ordered pair of points of X is t-apart in exactly λ blocks of B, then the (υ, k, λ)-MD is called a perfect design and is denoted briefly by (υ, k, λ)-PMD. In this paper, we shall be concerned mainly with the case where k=5 and λ>1. It will be shown that the necessary condition for the existence of a (υ, 5, λ)-PMD, namely, λv(υ-1)≡0 (mod 5), is also sufficient for λ>1 with the possible exception of pairs (υ, λ) where λ=5 and υ=18 and 28