Abstract
We introduce the Almost Intersection Property for pairs of possibly infinite matroids on the same groundset. We prove that if such a pair satisfies the Almost Intersection Property then it satisfies the Matroid Intersection Conjecture of Nash-Williams. We also present some corollaries of that result.
Highlights
The finitary case of the following Matroid Intersection Conjecture has been introduced by Nash-Williams
If M and N are matroids on a common groundset E, there exist disjoint I, J ⊆ E such that clM (I) ∪ clN (J) = E and I ∪ J is independent in both M and N
A covering for (M, N ) is a pair (A, B) of subsets of E that are independent in M, N, respectively, and A ∪ B = E
Summary
The finitary case of the following Matroid Intersection Conjecture has been introduced by Nash-Williams (see [1]). It is a generalization of the well known Edmonds’ Intersection Theorem. Property iff there exists a partition E = P C such that (M P , N P ) has a packing and (M.C, N.C) has a covering. Bowler and Carmesin [3] proved the following result showing that the Matroid Intersection Conjecture is equivalent to a conjecture involving the Packing/Covering Property. (M, N ) satisfies the Matroid Intersection Conjecture if and only if (M, N ∗) has the Packing/Covering Property. 2. (M, N ) has the Almost Packing/Covering Property This result has the following corollary: Corollary 1.6. If (M, N ) has the Packing/Covering Property and A, B ⊆ E are finite, (M, N ) /A\B has the Packing/Covering Property
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