Constructions and Uses of Incomplete Pairwise Balanced Designs

We give explicit constructions for incomplete pairwise balanced designs IPBD((v;w),K), or, equivalently, edge-decompositions of a difference of two cliques K_v ∖ K_w into cliques whose sizes belong to the set K. Our constructions produce such designs whenever v and w satisfy the usual divisibility conditions, have ratio v/w bounded away from the smallest value in K minus one, say v/w > k-1+ϵ, for k =min K and ϵ>0, and are sufficiently large (depending on K and ϵ). As a consequence, some new results are obtained on many related designs, including class-uniformly resolvable designs, incomplete mutually orthogonal latin squares, and group divisible designs. We also include several other applications that illustrate the power of using IPBDs as `templates'.