A finite embedding theorem for partial Steiner 3-designs

A Steiner system S(t, k, n) is a k-uniform set system. on [n] for which every t-set is covered exactly once. More generally, a partial Steiner. system P(t, k, n) is a k-uniform set system on [n] where every t-set is covered at most once. Let q be a prime power.. Using circle geometries and field-based block spreading, we give an explicit embedding for any partial Steiner system P(3, q + 1, n) into a Steiner system S(3, q + 1, q(m) + 1) for some m = m(q, n). (C) 2014 Elsevier Inc. All rights reseived.