On the existence of ( v,7,1)-perfect Mendelsohn designs

Abstract

Let v, k, and λ be positive integers. A ( v, k, λ)-Mendelsohn design (briefly ( v, k, λ)-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 λ of the blocks of B . If for all t = 1,2,…, k - 1, every ordered pair of points of X are t-apart in exactly λ of the blocks of B , then the ( v, k, λ)-MD is called a perfect design and denoted briefly by ( v, k, λ)-PMD. A necessary condition for the existence of a ( v, 7, 1)-PMD is v ≡ 0 or 1 (mod 7). We show that this condition is sufficient for all v ⩾ 2136, with at most 104 possible exceptions below this value. This result is established, for the most part, by means of a result on pairwise balanced designs (PBDs) which is of interest in its own right. If Q ∗ denotes the set of all prime powers congruent to 0 or 1 modulo 7, then it is shown that a PBD B( Q ∗, 1; v) exist for all integers v ⩾ 2136, where v ≡ 0 or 1 (mod 7), withat most 104 possible exceptions below this value.