On The Embeddings Of Resolvable Mendelsohn Triple System

A Mendelsohn triple systems of order v and indexλ, denotedMTS ( v ,λ), is a pair ( X ,Α)where X is a v- set andΑis acollection of cyclically ordered 3?subsets (called cyclic triples) ofX such that each ordered pair of distinct elements of X appears inpreciselyλcyclic triples ofΑ.An incomplete resolvable Mendelsohn triple system IRMTS (u , v )isa triple (Y , X ,Α), where Y is a u-set,X is a v-subset of Y,Αis acollection of cyclic triples of Y such that each ordered 2-subset of X iscontained in no cyclic triple ofΑ, each ordered pair of distinct elementsof Y , not both from X , is contained in precisely 1 cyclic triple ofΑ.In this thesis we studied the problem of the embeddings ofresolvable Mendelsohn Triple System, proving the following conclusion:Let u≡v≡0(mod3), IRMTS (u , v )exists if and only if u≥3vexcept possibly for 3 values where (u , v )∈{(150,48),(210,69),(261,84)}.