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We give explicit constructions for incomplete pairwise balanced designs IPBD((v; w), K), or, equivalently, edge-decompositions of a difference of two cliques $$K_v \setminus K_w$$Kv\Kw 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 $$v/w > k-1+\epsilon $$v/w>k-1+ϵ, for $$k =\min K$$k=minK and $$\epsilon >0$$ϵ>0, and are sufficiently large (depending on K and $$\epsilon $$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’.

Constructions and uses of incomplete pairwise balanced designs