Disjoint Bases for a Countable Family of Rank-Finite Matroids

Abstract

Let M = (Mr)r2R be a system of matroids on a set S. For every transfinite sequence f of distinct elements of S, we define a number (f). In (12) we proved that the condition that (f) 0 for every possible choice of f is necessary for M to have a system of mutually disjoint bases. Further, we showed that this condition is sucient if R is countable and Mr is a rank-finite transversal matroid for every r 2 R. In this paper, we prove that our condition is also sucient in the much more general case of countable systems of arbitrary rank-finite matroids.