A Necessary Condition for the Existence of Disjoint Bases of a Family of Infinite Matroids
Jerzy Wojciechowski · 1994
Let M = (Mr)r∈R be a system of matroids on a set S. Following the ideas of Nash-Williams [7], for every transfinite sequence f of distinct elements of S, we define a number η(f). We prove that the condition that η(f) ≥ 0 for every possible choice of f is necessary for M to have a system of mutually disjoint bases. Further, we show that this condition is sufficient if R is countable and Mr is a rank-finite transversal matroid for every r ∈ R. We also present conjectures about edge-disjoint spanning trees and detachments of countable graphs.