On the Matroid Intersection Conjecture

Abstract

In this dissertation, we investigate the Matroid Intersection Conjecture for pairs of matroids on the same ground set, proposed by Nash-Williams in 1990. Originally, the conjecture was stated for finitary matroids only, but we consider it for general matroids and introduce new approaches to attack the conjecture.;The first approach is to consider the situation when it is possible to make a finite modification to the matroids after which the pair satisfies the conjecture. In such a situation we say that the pair has the Almost Intersection Property. We prove that any pair of matroids with the Almost Intersection Property must satisfy the Matroid Intersection Conjecture. Using this result we prove that the Matroid Intersection Conjecture is true in the case when one of the matroids has finite rank and also in the case when one of the matroids is a patchwork matroid.;Our second new approach is inspired by the proof of the general version of Konig's Theorem for bipartite graphs. That result implies that the Matroid Intersection Conjecture is true for pairs of partition matroids. We develop some new techniques that generalize the critical set approach used in the proof of the countable version of Konig's Theorem. Our results enable us to prove that the Matroid Intersection Conjecture is true for a pair of singular matroids on a set that is infinitely countable. A matroid is singular when it is a direct sum of matroids such that each term of the sum is a uniform matroid either of rank one or of co-rank one.