A complete existence theory for Sarvate-Beam triple systems

An ordinary block design has v points, blocks of size k, and each of the ( v 2 ) unordered pairs of points is contained in a common number λ of blocks. In 2007, Sarvate and Beam introduced a variation where rather than covering all pairs a constant number of times, one wishes to cover all pairs a different number of times. It is of greatest interest when these pair frequencies form an interval of ( v 2 ) consecutive integers. The case k = 3 has received particular attention in the literature, and in this case the object in question is sometimes called a Sarvate-Beam triple system. We prove that the basic (counting) necessary condition v ≡ 0, 1 (mod 3) is sufficient for the existence of Sarvate-Beam triple systems, where any interval of consecutive pair frequencies is possible. Similar results are also obtained, with certain minor restrictions, when v ≡ 2 (mod 3).