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, {A i i ∈ I }, where I is a finite index set. |K| will denote the cardinality of a set K and I use the notation