Transversal Theory and Matroids

In this paper I use techniques developed by Mirsky and Perfect (5) to generalize the extremely close relationship between transversal theory and the theory of matroids or independence structures. I extend in two directions a fundamental theorem of Rado (8) and use the techniques of Mirsky and Perfect to obtain easy proofs of known and unknown results about systems of representatives with repetition.2. Basic concepts. In this section I review the results used subsequently. Throughout the paper, S will denote a finite set and A will denote the collection of subsets of S, {Ai i ∈ I}, where I is a finite index set. |K| will denote the cardinality of a set K and I use the notation