The spectrum for large sets of pure Mendelsohn triple systems

An LPMTS(v)LPMTS(v) is a collection of v-2v-2 disjoint pure Mendelsohn triple systems on the same set of vv elements. In this paper, the concept of t-purely partitionable Mendelsohn candelabra system (or t  -PPMCS in short) is introduced for constructing LPMTS(v)sLPMTS(v)s. A powerful recursive construction for t-PPMCSs is also displayed by utilizing s  -fan designs. Together with direct constructions, the existence of an LPMTS(v)LPMTS(v) for v≡1,9(mod12) and v>1v>1 is established. For odd integer v⩾7v⩾7, a special construction from both LPMTS(v)LPMTS(v) and OLPMTS(v)OLPMTS(v) to LPMTS(2v+1)LPMTS(2v+1) is set up. Finally, the existence of an LPMTS(v)LPMTS(v) is completely determined to be the set {v:v≡0,1(mod3),v⩾4,v≠6,7}.