On indecomposable pure Mendelsohn triple systems

AbstractLet v and λ be positive integers. A Mendelsohn triple system MTS(v, λ) is a pair (X, B), where X is a v-set (of points) and B is a collection of cyclically ordered 3-subsets of X (called blocks or triples) such that every ordered pair of points of X is contained in exactly λ blocks of B. If we ignore the cyclic order of the blocks, then an MTS(v, λ) can be viewed as a (v, 3, 2λ)-balanced incomplete block design (BIBD). An MTS(v, λ) is called pure if its underlying (v, 3, 2λ)-BIBD contains no repeated blocks. An MTS(v, λ) is indecomposable if it is not the union of two Mendelsohn triple systems MTS(v, λ1) and MTS(v, λ2) with λ = λ1 + λ2. For λ = 2 and 3, the problem of existence of a pure MTS(v, λ) is completely solved in this paper. We further prove that an indecomposable pure MTS(v, 2) exists if and only if v = 0 or 1 (mod 3) and v ⩾ 6, except possibly v = 9, 12. We show that an indecomposable pure MTS(v, 3) exists for v = 8, 11, 14 and for all v ⩾ 17