Large Sets Of Combinatorial Designs

A cyclic triple (x, y, z) on X is a set of three ordered pairs (x, y), (y, z) and (z, x) of X. A transitive triple (x,y,z) on X is a set of three ordered pairs (x,y), (y,z) and (x, z) of X. An oriented triple system of order v is a pair (X, β) where X is a v-set and β is a collection of cyclic or transitive triples on X, called blocks, such that every ordered pair of X belongs to exactly one block of B. In particular, if the triples in β are all cyclic (or transitive), then (X, β) is called a Mendelsohn (or directed) triple system and denoted by MTS(v) (or DTS(v)).An MTS(v) (X, A) is called pure and denoted by PMTS(v) if (a, b, c) ∈ A implies (c,b,a) (?) A. Similarly, A pure DTS(v) (PDTS(v) in short) (X,A) is a DTS(v) in which (a, b, c) ∈ A implies (c, b, a) (?) A.An MTS(v) (or a DTS(v)) is called resolvable if its blocks can be partitioned into subsets (called parallel classes), each containing every element of X exactly once. A resolvable MTS(v) and a resolvable DTS(v) are denoted by RMTS(v) and RDTS(v) respectively.A large set of MTS(v)s (or DTS(v)s), denoted by LMTS(v) (or LDTS(v)), is a collection of v - 2 MTS(v)s (or 3(v - 2) DTS(v)s) on X such that every cyclic triple (or transitive triple) on X occurs as a block in exactly one of the v — 2 MTS(v)s (or 3(v - 2) DTS(v)s). An LPMTS(v) (or LPDTS(v)) denotes an LMTS(v) (or LDTS(v)) in which each MTS(v) (or DTS(v)) is pure. An LRMTS(v) (or LRDTS(v)) denotes an LMTS(v) (or LDTS(v)) in which each MTS(v) (or DTS(v)) is resolvable.An overlarge set of MTS(v), denoted by OLMTS(v), is a collection {(X \...