Intersections among Steiner systems

A Steiner system S(l, m, n) is a system of subsets of size m (called blocks) from an n-set S, such that each d-subset from S is contained in precisely one block. Two Steiner systems have intersection k if they share exactly k blocks. The possible intersections among S(5, 6, 12)'s, among S(4, 5, 11)'s, among S(3, 4, 10)'s, and among S(2, 3, 9)'s are determined, together with associated orbits under the action of the automorphism group of an initial Steiner system. The following are results: (i) the maximal number of mutually disjoint S(5, 6, 12)'s is two and any two such pairs are isomorphic; (ii) the maximal number of mutually disjoint S(4, 5, 11)'s is two and any two such pairs are isomorphic; (iii) the maximal number of mutually disjoint S(3, 4, 10)'s is five and any two such sets of five are isomorphic; (iv) a result due to Bays in 1917 that there are exactly two non-isomorphic ways to partition all 3-subsets of a 9-set into seven mutually disjoint S(2, 3, 9)'s.