Proof of Nash-Williams' Intersection Conjecture for countable matroids

TL;DR

This paper proves Nash-Williams' Matroid Intersection Conjecture for countable finitary matroids, establishing the existence of a common independent set with spanning properties in a countable setting.

Contribution

It provides the first positive proof of Nash-Williams' conjecture for countable matroids, expanding the understanding of matroid intersection theory.

Findings

Confirmed the conjecture for countable finitary matroids

Constructed a common independent set with spanning properties

Extended matroid intersection results to infinite, countable cases

Abstract

We prove that if $ M $ and $ N $ are finitary matroids on a common countable edge set $ E $ then they admit a common independent set $I $ such that there is a bipartition $ E=E_{M}\cup E_{N} $ for which $ I\cap E_M $ spans $ E_M $ in $ M $ and $ I\cap E_N $ spans $ E_N $ in $ N $. It answers positively the Matroid Intersection Conjecture of Nash-Williams in the countable case.