Problems and results on block designs and set systems

Let J SJ=n . A family F of subsets A i c S , 1<-kgn , is called an r -design if every r -tuple {x 1 , . . ., xr } of S is contained in one and only one of the A 's . If r=2 we will just call these systems designs . (In the literature these 2 -designs are usually called partially balanced block designs .) Usually we will restrict ourselves to designs ; i .e ., to r=2 . There are two rivial designs . The design with m=1, IA I J=n is completely trivial and will henceforth be ignored . Almost as trivial is the r -design with m=,+(n_ 1 ) HA l l=n-1, A i l=r, 2sis1+(n_ 1 ) . For r=2 this trivial design is sometimes called the near pencil . Henceforth, i-t will also be ignored .