A recursive construction for regular difference triangle sets

A difference triangle set (DΔS) is a collection of sets of integers having the property that every integer can be written in at most one way as the difference of two elements within a set of the collection. The standard objective is to minimize the largest difference represented, given a specified size of the collection and sizes of the sets that it contains. In order to construct DΔSs, we present a new type of combinatorial design, monotonic directed (v, k, λ)-designs (MDDs). Using MDDs, we give a general recursive construction for difference triangle sets (DΔSs). Several instances of this main construction are derived. One of these, the perfect construction, leads to an infinite family of regular (optimal) DΔSs if the existence of a single regular DΔS is known.