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 ) .