Transversal matroids, not
necessarily having finite character, are investigated. It is demonstrated that if
U(I) = (Ai: i ∈ I) is an arbitrary family of subsets of an arbitrary set E whose
transversal matroid has at least one basis and has no coloops, then A(I) has a
transversal; in fact, each basis is a transversal of A(I) but of no proper
subfamily of A(I). P. Hall’s theorem on the existence of a transversal for
a finite family, and indeed an extension of it, can be obtained from this
result.
Some necessary conditions for a matroid to be a transversal matroid are derived.
One of these is that a transversal matroid of rank r can have at most k-flats
having no coloops (1 ≦ k ≦ r).
