On Graphical Quintuple Systems

It is well-known that the following divisibility conditions are necessary for the existence of a t-(v, k, A) design: A(V-~) == 0 mod (k~), Os; i:::; t. t-, t-, A t.( v, k, A) design (0, £ilJ) with £ilJ =0 or ffi=Lk(n) is said to be trivial. One can show by elementary counting arguments that in a trivial t-(v, k,A) design, we must have either A=0 or A = (k=:)' The complement of a t.(v, k, A) design (0, £ilJ) is the ordered pair (0, Lk(O)\£ilJ). It is easy to show that the complement of a t-(v, k, A) design is a t· (v, k, (k=:) A) design. A (k -1)-( v, k, A) design is also commonly called a k-tuple system. Let 0 be the set of v =mlabelled edges of the undirected complete graph K p • An ordered pair (0, £ilJ) is a graphical t-(v, k, A) design if