Maximal consistent families of triples

The general problem of dividing a set into a maximal collection of subsets in such a way that the subsets will overlap in certain specified ways is a fundamental problem in experimental design and in the design of systems involving shared use of system elements. This memorandum solves this problem in the case for which the subsets are each required to have three elements, and any pair of subsets have at most one element in common. A family F of three element subsets of an n-element set S_n is called n-consistent if the intersection of any two sets of F contain at most one element of S_n. We find maximal (in number of elements) F for all n. For certain n the F are Steiner Triples Systems. The construction of the F is constructive. Structure Theorems are given determining the graph of doublets not covered by triplets in F.