Resolving Triple Systems into Regular Configurations

A λ − Triple System(v), or a λ–TS(V,B), is a pair (V, B) where V is a set and B is a subset of the 3-subsets of V so that every pair is in exactly λ elements of B. A regular configuration on p points with regularity ρ on l blocks is a pair (P,L) where L is a collection of 3-subsets of a (usually small) set P so that every p in P is in exactly ρ elements of L, and |L| = l. The Pasch configuration ({0, 1, 2, 3, 4, 5}, {012, 035, 245, 134}) has p=6, l=4, and ρ=2. A λ–TS(V,B), is resolvable into a regular configuration C=(P,L), or C–resolvable, if B can be partitioned into sets Πi so that for each i, (V,Πi) is isomorphic to a set of vertex disjoint copies of (P,L). If the configuration is a single block on three points this corresponds to ordinary resolvability of a Triple System. In this paper we examine all regular configurations C on 6 or fewer blocks and construct C–resolvable λ–Triple Systems of order v for many values of v and λ. These conditions are also sufficient for each C having 4 blocks or fewer. For example for the Pasch configuration λ ≡ 0 (mod 4) and v ≡ 0 (mod 6) are necessary and sufficient. MRSC #05B07