Uniqueness of an independent transversal of a family of sets

Abstract

In his book Transversal Theory (4) Mirsky introduces the problem of finding a necessary and sufficient condition in order that a finite family of sets have a unique independent transversal. The purpose of this note is to give one necessary and sufficient condition. More generally, we give a necessary and sufficient condition in order that a common basis of two rank-finite matroids be unique. We then specialize this result to give an answer to the above question.