The notion and basic properties of M-transversals

AbstractLet I be a finite index set and let A denote the family (Ai : i ∈ I) of finite subsets of S. Let M be a matroid without loops on I. A family (xi : i ∈ I) of elements of S is an M-system of representatives of A if xi> ∈ Ai, for any i ∈ I, and the set {i ∈ I : xi = s} is independent in M, for any s ∈ S. Let (xi : i ∈ I) be an M-system of representatives of A; then the set X = {xi:i I} (i.e., the set of distinct elements of the system (xi : i ∈ I)) is called the M-transversal of A. (if Uk is the k-uniform matroid of rank k, then the Uk-transversal is usually described as k-transversal, or as system of presentatives with repetition.) The aim of this note is to prove an M-transversal. version of Rado's and Perfect's Theorem and to give a short proof of a similar result known for k-transversals