E. C. Milner

TL;DR: This paper proves a conjecture of C. St. J. Nash-Williams giving necessary and sufficient conditions for an arbitrary countable system of sets to have a transversal.

TL;DR: In this article, it was shown that the Nash-Williams theorem does not necessarily imply finite matchability for matroids, and the authors gave a simple example (Example 3.2.2) to illustrate this.

TL;DR: It is shown that, if a matroid is f.m.e., then it has a perfect matching, which generalizes a well-known theorem of Nash-Williams which states that the edges of a graph can be covered by edge-disjoint circuits if every finite set of edges can be so covered.

TL;DR: It is shown that a necessary and sufficient condition for the existence of an independent transversal of a countable system F is that Fi does not depend upon F(K) whenever F K is an ℳ-critical subsystem and i∈I\K.