On the intersection of infinite matroids

Abstract

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved recently by Aharoni and Berger.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays. * Research supported by the Minerva foundation. 1 see Theorem 3.1 below.

Topological cycle matroids of infinite graphs

“…A tame matroid is binary if every circuit and cocircuit always intersect in an even number of edges. 1 Roughly, a binary presentation of a tame matroid M is something like a pair of representations over F 2 , one of M and of the dual of M , formally: Definition 2.7. Let E be any set.…”

On the Matroid Intersection Conjecture

“…As a motivation for working on this conjecture, we can point to the infinite Menger theorem. The infinite Menger theorem was conjectured by Erdős in the 1960s and proved recently by Aharoni and Berger [7]. It states that if A and B are sets of vertices in a (possibly infinite) graph G, then there exists a family P of disjoint A− B−paths and a separating set which consists of exactly one vertex from each path in P. Due to the complexity of the only known proof of this theorem, the investigation of a matroidal proof of the infinite Menger theorem attracts attention among researchers.…”

“…It states that if A and B are sets of vertices in a (possibly infinite) graph G, then there exists a family P of disjoint A− B−paths and a separating set which consists of exactly one vertex from each path in P. Due to the complexity of the only known proof of this theorem, the investigation of a matroidal proof of the infinite Menger theorem attracts attention among researchers. In [7] this is shown; specifically, it is proved that the Matroid Intersection Conjecture 1. • When M is finitary and N is a countable direct sum of finite rank matroids ( [5]).…”

“…• When M is finitary and N is co-finitary ( [7]). (We call a matroid finitary if all its circuits are finite and co-finitary if its dual is finitary).…”

“…• When M is nearly finitary and N is the dual of a nearly finitary matroid ( [7]). (We call a matroid nearly finitary if by removing finitely many elements from any subset that contains no finite circuit, we get an independent set).…”