On generalised -designs and their parameters

Recently, P.J. Cameron studied a class of block designs which generalises the classes of t -designs, α -resolved 2-designs, orthogonal arrays, and other classes of combinatorial designs. In fact, Cameron’s generalisation of t -designs (when there are no repeated blocks) is essentially a special case of the “poset t -designs” in product association schemes studied ten years earlier by W.J. Martin, who further studied the special case of “mixed block designs”. In this paper, we study Cameron’s generalisation of t -designs from the point of view of classical t -design theory, in particular investigating the parameters of these generalised t -designs. We show that the t -design constants λ i (the number of blocks containing an i -subset of the points, where i ≤ t ) and λ i j (the number of blocks containing an i -subset I of the points and disjoint from a j -subset J of the points, where I ∩ J = 0̸ and i + j ≤ t ) have very natural counterparts for generalised t -designs. Our main result places strong restrictions on the block structure of Cameron’s t - ( v , k , λ ) designs, an important subclass of generalised t -designs. We also generalise N.S. Mendelsohn’s concept of “intersection numbers of order r ” for t -designs, and show that analogous equations to those of Mendelsohn hold for generalised t -designs.