Restricted Simple \(1\)-Designs

Given a partition \(\{P_1, \ldots, P_m\}\) of a \(v\)-set, a restricted simple \(1\)-design is a collection of distinct subsets (blocks) such that every element occurs in the same number of blocks, but any two elements from the same part do not occur together in the same block. We give a construction of restricted simple \(1\)-designs to show that the necessary conditions are sufficient for the existence of restricted simple \(1\)-designs.