The Almost Intersection Property for Pairs of Matroids on Common Groundset

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 NashWilliams. We also present some corollaries of that result. Mathematics Subject Classifications: 05B35


Introduction
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. We say that the pair (M, N ) has the Almost Intersection Property when there exist almost disjoint I, J ⊆ E such that cl M (I)∪cl N (J) is almost equal to E and I ∪J is almost independent in both M and N . We mean here that the sets I ∩ J, E (cl M (I) ∪ cl N (J)) and (I ∪ J) K are all finite for some K ⊆ E that is independent in both M and N .
We say that (S, T ) is an almost packing of (M, N ) iff S and T are spanning in M and N , respectively, and S ∩ T is finite. Analogously, we say that (I, J) is an almost covering of (M, N ) iff I and J are independent in M and N , respectively and E (I ∪ J) is finite. If there exists a partition E = P Q of E such that (M \Q, N \Q) has an almost packing and (M/P, N/P ) has an almost covering, then we say that (M, N ) has the Almost Packing/Covering Property.
If F ⊆ E, then say that (M, N ) has the Packing/Covering Property modulo F iff there exists a partition E F = P C such that (M, N ) /F \C has a packing and (M, N ) \F/P has a covering. The following proposition will be proved in the next section. The main result of this paper is the following theorem.

Preliminaries
We follow the notation and terminology of [6] and [5]. In particular, the following statements about a set I ⊆ P (E) are independence axioms: (I2) I is closed under taking subsets. Proof of Proposition 1.3. Assume first that there exists a partition E = P Q F such that F is finite, (M, N * ) /F \Q has a packing and (M, N * ) \F/P has a covering. Then P and Q can be partitioned as P = S T and Moreover, we can assume without loss of generality that S is independent in M and B is independent in N (see Since Now assume that (M, N ) has the Almost Intersection Property. Let I, J ⊆ E be almost disjoint and such that cl M (I) ∪ cl N (J) is almost equal to E and I ∪ J is almost independent in both M and N . Without loss of generality, we can assume that I is independent in M and J is independent in N . Let I ⊆ I J and J ⊆ J I be such that I ∪ J is a basis of M I∪J and J ∪ I is a basis of N I∪J . Note that ( Figure 2.2). Note that F is finite and P ⊆ cl M (I ∪ F ). Moreover, since I ∪ J is independent in N and J spans every element of Q, it follows that I is independent in N/Q and hence P ∪ F is spanning in (N/Q) * = N * \Q. It follows that (I , P ) is a packing for (M, N * ) /F \Q. Similarly, (J , Q ) is a packing for (N, M * ) /F \P and hence it is a covering for (M, N * ) \F/P .

The Main Result
Throughout this section we assume that M and N are matroids on a common ground set E.
A • if i is odd, then there exists a circuit C of M with e i , e i+1 ∈ C ⊆ B ∪ {e i }; • if i is even, then there exists a circuit C of N with e i , e i+1 ∈ C ⊆ D ∪ {e i }.
We say that such a chain is from e 1 to e n .
A (B, D)-exchange N -chain is defined analogously with the words "even" and "odd" interchanged. A (B, D)-exchange chain refers to either of these notions.
The following lemma is proved in [2]. The following lemma is the key technical result that will be used in the proof of the main result. The proof of the following lemma is routine.  Proof of Theorem 1.5. It is clear that 1. implies 2., which implies 3. It suffices to show that 3. implies 1.
Assume that (M, N ) has the Packing/Covering Property modulo finite F ⊆ E. Let P Q be a partition of E F such that (M, N ) /F \Q has a packing and (M, N ) \F/P has a covering. Since F is finite, it follows that (M \Q, N \Q) has an almost packing and hence it has a semi-packing. Consequently, Lemma 3.3 implies that (M \Q, N \Q) has the Packing/Covering Property. Let E Q = P Q be a partition of E Q such that (M P , N P ) has a packing (S, T ) and (M, N ) \Q/P has a covering (A, B) (see Figure 3.2). A covering of (M, N ) \F/P is an almost covering of (M/P, N/P ) and P P is finite so Corollary 3.6 implies that (M/ (P ∪ P ) , N/ (P ∪ P )) has an almost covering (I, J). Since F is finite, it follows that ((A ∪ I) F, (B ∪ J) F ) is an almost covering of (M/P , N/P ). Since (M/P , N/P ) has a semi-covering, Corollary 3.4 implies that it has the Packing/Covering Property.
Let P Q be a partition of E P such that (M, N ) /P \Q has a packing (S , T ) and (M, N ) /P /P has a covering (A , B ) (see Figure 3.3).

Consequences
Our results imply the following strengthening of Edmonds' Intersection Theorem. The following result is proved in [2]. In [2] it is proved that if M ∨ N * is a matroid, then (M, N ) satisfies the Intersection Conjecture. In particular, the following result holds. We can use Corollary 3.4 to provide an alternative proof. In [4] "patchwork matroids" are introduced and proved to satisfy the following characterization. Here K B = (K B) ∪ (B K).