On indecomposable pure Mendelsohn triple systems

Abstract Let 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 , λ 1 ) and MTS( v , λ 2 ) with λ = λ 1 + λ 2 . 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.