Generalizations of Cole’s Systems

There are four resolvable Steiner triple systems on fifteen elements. Some generalizations of these systems are presented here. 0. Introduction. The following definition is standard (see [2]). Definition. Let V be a finite set, v = #(V ) be the number of elements of V , let (V/3) (resp. (V/2)) be the set of all unordered triads (resp. pairs) of distinct elements of V . A Steiner triple system on V (or of order v) is a subset S of (V/3) such that any pair from (V/2) is contained in one and only one triad of S. Such a system is said to be resolvable if there exists a partition (1) S = S1 ∪ S2 ∪ . . . ∪ Sn of S into n disjoint subsets Si, where n = (v − 1)/2, #(Si) = v/3, all the triads inside each Si are disjoint (i.e. each Si presents a partition of V into disjoint triads), i = 1, 2, . . . , n. Such a partition (1) is said to be Kirkman structure (or resolution) on the set V . Frank Nelson Cole found in [1] four resolvable Steiner triple systems of order 15. Any resolvable system of order 15 is isomorphic to one and only one of these Cole’s 1991 Mathematics Subject Classification: 05B05, 05B30, 20K01