First order convergence of matroids^{†}^{†}thanks: Research supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007 2013)/ERC grant agreement no. 259385. The work of the second author was also supported by the Engineering and Physical Sciences Research Council Standard Grant number EP/M025365/1.
Abstract
The model theory based notion of the first order convergence unifies the notions of the leftconvergence for dense structures and the BenjaminiSchramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded treedepth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branchdepth representable over a fixed finite field has a limit modeling, i.e., there exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties. We show that neither of the bounded branchdepth assumption nor the representability assumption can be removed.
1 Introduction
The theory of combinatorial limits keeps attracting a growing amount of attention. Combinatorial limits have sparked many exciting developments in extremal combinatorics, in theoretical computer science, and other areas. Their significance is also evidenced by a recent monograph of Lovász [25]. The better understood case of convergence of dense structures originated in the series of papers by Borgs, Chayes, Lovász, Sós, Szegedy, and Vesztergombi [6, 7, 8, 26, 27] on the dense graph convergence, and the notion was applied in various settings including hypergraphs, partial orders, permutations, and tournaments [12, 15, 19, 20, 21, 24]. The convergence of sparse structures (such as graphs with bounded maximum degree) referred to as the BenjaminiSchramm convergence [1, 2, 11, 14] is less understood despite having links to many problems of high importance. For example, the conjecture of Aldous and Lyons [1] on BenjaminiSchramm convergent sequences of graphs is essentially equivalent to Gromov’s question of whether all countable discrete groups are sofic. Other notions of convergence of sparse graphs were also proposed and studied [3, 4, 5, 11, 14].
In the light of many results on the convergence of graphs, one can ask whether a reasonable theory of matroid convergence can be developed. The first obstacle to building such a theory comes from the fact that matroids when viewed as hypergraphs (e.g. with edges being the bases) can be too sparse for the classical dense convergence approach to be directly applied, and too dense for the sparse convergence approach at the same time. For example, the number of bases of the graphic matroid of is , an exponentially small fraction of all element subsets of the edge set of and an even tinier fraction of all subsets of the edge set, which rules out the dense convergence approach. On the other hand, each element of this matroid is contained in a nonconstant number of bases, and it is impossible to follow the sparse convergence approach. We overcome this obstacle by adapting the notion of the first order convergence to matroids.
The notion of the first order convergence was introduced by Nešetřil and Ossona de Mendez [28, 29] as an attempt to unify the convergence notions in the dense and sparse settings: a sequence of structures of a fixed type (e.g., graphs) is first order convergent if the probability that a random tuple of its elements has a first order property , converges for every choice of (a formal definition can be found in Subsection 2.4). It holds that every first order convergent sequence of dense structures is convergent in the dense sense and every first order convergent sequence of sparse structures is convergent in the BenjaminiSchramm sense.
In the analogy to graphons in the setting of dense graphs and graphings in the setting of sparse graphs, an analytic limit object called a limit modeling was proposed in [28, 29] to represent asymptotic properties of first order convergent sequences. Unlike in the dense and sparse graph settings, it is not true that every first order convergent sequence of graphs has a limit modeling. For example, the sequence of ErdősRényi random graphs for is first order convergent with probability one but it has no limit modeling [29, Lemma 18]. In the same paper, Nešetřil and Ossona de Mendez showed the following.
Theorem 1.1.
Every first order convergent sequence of graphs with bounded treedepth has a limit modeling.
This result was extended to first convergent sequences of trees and graphs of bounded pathwidth [23, 30]. Nešetřil and Ossona de Mendez [31] have recently shown that every first order convergent sequence of graphs from a nowheredense class of graphs has a limit modeling, which is the most general result possible for monotone classes of graphs [29, Theorem 25].
As a test that the approach to the matroid convergence based on the first order convergence is meaningful, it seems natural to prove the analogue of Theorem 1.1, which is actually one of our results (Theorem 1.2). On the way towards Theorem 1.2, we need to find a matroid parameter that can play the role of the graph treedepth. We do so by introducing a parameter called branchdepth in Section 3. We believe that this matroid parameter is the right analogue of the graph treedepth because it has the following properties, which we establish in this paper. We refer the reader to [32, Chapter 6] for a thorough discussion of the graph treedepth.

The branchdepth of a matroid corresponding to a graph is at most the treedepth of .

The branchdepth of a matroid corresponding to a graph with treedepth is at least if is connected.

The branchdepth is a minor monotone parameter (the same holds for graph treedepth).

The branchdepth of a matroid is at most the square of the length of its longest circuit (recall that the treedepth of a graph is at most the length of its longest path).

The branchdepth of a matroid is at least the binary logarithm of the length of its longest circuit (recall that the treedepth of a graph is at least the binary logarithm of the length of its longest path).
In addition, there exists an efficient algorithm that given an integer and an oraclerepresented input matroid either outputs its decomposition of bounded depth or it determines that the branchdepth of the input matroid exceeds .
Equipped with the notion of branchdepth, we prove the following theorem in Section 4.
Theorem 1.2.
Every first order convergent sequence of matroids with bounded branchdepth that is representable over a fixed finite field has a limit matroid modeling.
Note that matroids representable over finite fields have a structure more similar to graphs than general matroids and it is not surprising that Theorem 1.2 includes this assumption. In fact, we show in Section 5 that neither the assumption on the bounded branchdepth nor the assumption on the representability over a fixed finite field can be dropped. In particular, we construct a first order convergent sequence of binary matroids that has no limit modeling, and a first order convergent sequence of rank three matroids representable over rationals that has no limit modeling.
2 Notation
In this section, we introduce the notation used throughout the paper.
2.1 Finite matroids
We start by introducing concepts related to finite matroids. We refer to the monograph by Oxley [34] for a more detailed treatment. In Subsection 2.3, we extend the terminology to infinite matroids. A matroid is a pair where is a finite set, called the ground set, and is a collection of its subsets referred to as independent sets. The set is required to be nonempty, to be hereditary (i.e., for every , must contain every subset of ), and to satisfy the augmentation axiom: if and are independent sets with , then there exists such that . We abuse the notation and we often denote by the ground set of the matroid .
A subset is called dependent if , and a minimal dependent set is a circuit. The number of elements of a circuit is referred to as its length. It is wellknown that the collection of circuits of a matroid satisfies the following properties.

.

If and , then .

If with and , then there exists a circuit such that .
Furthermore, (C1)–(C3) form an alternative set of axioms to define matroids. More precisely, a collection of subsets of is the collection of circuits of a matroid if and only if it satisfies (C1)–(C3). The rank of a set is the size of the largest independent subset of . The rank of a matroid is the rank of the ground set of . It is wellknown that the rank function of a matroid is submodular, i.e., for any two subsets it holds that . When there is no danger of confusion, we omit the subscript, i.e., we just use instead of .
There are two particular important examples of matroids. A graphic matroid is obtained from a graph in the following way: the elements of are the edges of , and a set of edges is independent if it is acyclic. Vector matroids have a set of vectors of a vector space as their ground set, and a set of elements is independent if they are linearly independent.
A matroid is called representable over a field if there exists a function that maps the elements of to vectors over such that is independent in if and only if is linearly independent. A matroid is binary iff it is representable over the binary field .
A loop is an element of with , and a bridge is an element such that . Two elements and are parallel if neither of them is a loop and . If is a subset of elements of , the closure of is defined as Clearly, .
If is a subset of the elements of , then is the matroid obtained from by deleting the elements of , i.e., the elements of are those not contained in , and a set is independent in iff it is independent in . The matroid is obtained by contracting : the elements of are those not contained in , and a subset of such elements is independent in iff is independent in and . When is a single element, we write and instead of and . The restriction of a matroid to is the matroid , where denotes the complement of in . Finally, a minor of a matroid is a matroid obtained by a sequence of deleting and contracting some of its elements. It is not hard to show that if a graph is a minor of a graph , then the matroid is a minor of the matroid .
A matroid is connected if the only two subsets satisfying are the empty set and the whole ground set. A component of is a set that is an inclusionwise maximal subset such that is connected. The components of are equivalence classes given by the binary relation that represents that two elements of are contained in a common circuit. Hence, any two components of a matroid are disjoint. If is a matroid and are its components, then .
2.2 Matroid algorithms
Algorithms for matroids have been studied extensively, and we want to review selected important facts here. It is common (see, e.g., [13, 35, 36]) to assume that the input matroid is presented by means of an independence oracle. That is, we assume that we can determine whether any subset of the elements of the given matroid is independent using a blackbox function in unit time. The complexity of algorithms for matroids is measured in terms of the number of elements of the input matroid.
There is an efficient algorithm [37] to test whether a given binary matroid is graphic, and if so to find a suitable graph. However, in general, deciding if an oraclegiven matroid is binary cannot be solved in subexponential time [36]. In the same paper [36], Seymour presents a polynomialtime algorithm to decide whether an oraclegiven matroid is graphic.
A lot of decision problems for matroids involve the structural matroid parameter branchwidth. Rather than giving the exact definition here, let us just say that matroid branchwidth is the analogue of graph treewidth. Oum and Seymour [33] showed, improving [16], that for every fixed , it can be decided in polynomial time whether the branchwidth of an oraclegiven matroid is at most , and that an optimal branchdecomposition can be constructed (for such matroids).
If is a class of matroids that are representable over a fixed finite field and that have branchwidth bounded by a constant, then properties expressible in monadic second order over can be decided in cubic time [17]. In [13], Gavenčiak, Oum, and the second author introduce the notion of locally bounded branchwidth and present a fixed parameter algorithm to decide first order properties on the class of regular matroids with locally bounded branchwidth.
Hliněný [18] also showed that for every field of order at least four, it is NPhard to decide whether a matroid given by its rational representation is representable. The result still holds when restricting the input to matroids of branchwidth at most three. On the other hand, for every and any two finite fields and , there is a polynomialtime algorithm that decides whether a given representable matroid of branchwidth at most is also representable [22].
Similarly to the relation between the treedepth and treewidth, the branchwidth of a matroid is upper bounded by the branchdepth of up to an additive constant. It is natural to ask whether Theorem 1.2 can be extended to sequences of matroids representable over finite fields that have bounded branchwidth; we believe that this is likely to be the case but it might be challenging to prove since the analogous statement for sequences of graphs has been proven only very recently [31].
2.3 Infinite matroids
One of the ways to define the notion of infinite matroids is to require the augmentation axiom to hold for finite subsets and to additionally require that an infinite set is independent if and only if all of its finite subsets are independent. Such matroids are called finitary. The drawback of this definition is that finitary matroids can have only finite circuits.
A robust notion of infinite matroids was proposed by Bruhn et al. [9]. They developed five equivalent axiom systems that characterize (infinite) matroids through independent sets, bases, the closure operator, circuits, and their rank function. We present the characterization through circuits here. Let be a set, and let be a collection of subsets. Further, let denote the independent sets, that is the collection of sets such that for all . A family is the collection of circuits of a matroid if it satisfies (C1), (C2), and the following two conditions.

Whenever and is a family of elements of such that for all , then for every there exists such that .

Whenever and , the set has a maximal element.
If , the axiom (C3’) becomes the usual strong circuit elimination axiom (C3). We remark that all finitary matroids are matroids in the sense just defined.
2.4 First order convergence
For a set of relational symbols, let denote the set of first order formulas using symbols from , and let denote the set of all such formulas with free variables. A modeling (or just a modeling if is clear from the context) is a (finite or infinite) structure whose domain is a standard Borel space equipped with a probability measure such that the following holds: for every , the subset formed by all tuples of the elements of satisfying is measurable with respect to the product measure .
For a formula and a modeling , the Stone pairing is , i.e., the Stone pairing is the probability that a randomly chosen tuple of the elements of satisfies . When a finite structure with elements is viewed as a modeling with a uniform discrete probability measure, it holds that
A sequence of finite structures is first order convergent if the sequence converges for every first order formula . A modeling is a limit modeling of a first order convergent sequence if
for every formula .
In Sections 3 and 4, we work with rooted trees, rooted forests and their modelings, which were studied in [29, Part 3]. A rooted tree is a tree with a distinguished vertex referred to as the root, and a rooted forest is a graph such that each of its component is a rooted tree. The depth of a rooted tree is the length of the longest path from the root to a leaf, and the depth of a rooted forest is the maximum depth of a rooted tree contained in it. Rooted forests can be described by a language with a single binary relation representing the parentchild relation. In addition, we also consider rooted forests with vertices colored with one of a bounded number of colors. The vertex coloring of a rooted forest that uses colors can be described by extending the language with unary relations, each representing one of the colors. We use the following [29, Theorem 34] to prove one of our main results.
Theorem 2.1.
Let and be fixed integers. Every first order convergent sequence of rooted forests with depth at most and with vertices colored with at most colors has a limit modeling.
Theorem 2.1 is proven in [29] for rooted forests described by a language with a single symmetric binary relation representing edges and a single unary relation distinguishing roots of trees in the forest. Since there is a basic interpretation scheme translating the description of rooted forests from this language to the language that we consider here and there is also a basic interpretation scheme in the other direction (see the next subsection for a definition if needed), Theorem 34 from [29] and Theorem 2.1 are equivalent by [29, Propositions 3 and 4].
Our paper concerns matroids and their modelings; we now introduce the notation related to the first order convergence matroids and matroid modelings. Let be the countable language containing a ary relation for every positive integer . The relation is formed by all tuples of elements that are independent in the matroid. Finite matroids can be axiomatized in the first order language by a countable set of formulas. However, in general, the axioms (C1), (C2), (C3’) and (CM) cannot be replaced by a countable set of first order axioms.
Let be a sequence of finite matroids, equipped with the uniform measure on its element sets. We define to be first order convergent if the sequence of the Stone pairings converges for every first order formula . A modeling is a limit modeling of if it is an infinite matroid and
for every first order formula . Note that this definition is stronger than that of a limit modeling because we require additionally that the limit modeling is an infinite matroid. Note that if there exists an integer such that every circuit of has length at most , then every circuit of the limit modeling has length at most . In particular, is finitary.
2.5 Interpretation schemes
Let be signatures, where has relational symbols with respective arities . An interpretation scheme of structures in structures is defined by an integer , which is called the exponent of the interpretation scheme, a formula , a formula , and a formula for each symbol , such that:

the formula defines an equivalence relation on tuples;

each formula is compatible with , in the sense that for every it holds
where , and represent tuples of free variables, and stands for .
For a structure , we denote by the structure defined as follows:

the domain of is the subset of the equivalence classes of the tuples such that ;

for each and every such that for every it holds
If is a tautology and is the equality on tuples, then the interpretation scheme is said to be basic.
The following is a standard result.
Proposition 2.2.
Let be an interpretation scheme of structures in structures. Then there is a mapping (defined by means of the formulas above) such that for every , and every structure , the following property holds. For every (where ) it holds
We need the following generalization of Propositions 3 and 4 from [29].
Lemma 2.3.
Let be an interpretation scheme of structures in structures, let be the mapping from Proposition 2.2, and let be a sequence of finite structures such that

, and

for every .
If the sequence is first order convergent, then the sequence is also first order convergent. Moreover, if is a limit modeling of , then is a limit modeling of .
Proof.
The sequence is first order convergent by [29, Proposition 3]. Let be the underlying algebra of the probability space of . We define the algebra on as follows:
The probability measure on is defined as for , where is the probability measure on . The condition (1) implies that is indeed a probability measure, i.e., .
We now argue that is a limit modeling of the sequence . Let . Proposition 2.2 yields that iff . It follows from the definition of the algebra that is measurable. The definition of yields that that
which combines with the condition (2) to the following:
Hence, is a limit modeling of the sequence . ∎
3 Matroid branchdepth
In this section, we introduce a matroid parameter analogous to the graph treedepth. We also present an algorithm that efficiently computes an approximate value of the parameter of an input matroid together with the certifying depthdecomposition.
3.1 Definition and basic properties
The branchdepth of a matroid is equal to the optimal height of a certain kind of a decomposition tree. In the definition below and in the proofs of subsequent claims, we use to denote the number of edges of a tree .
Definition 3.1.
Let be a finite matroid. A depthdecomposition of is a pair , where is a rooted tree and is a mapping such that

, and

for every ,
where is the union of paths from the root to all the vertices in . The branchdepth of a matroid , denoted by , is the smallest depth of its depthdecomposition, i.e., the smallest depth of a rooted tree such that is a depthdecomposition of .
For any matroid there is a trivial decomposition where the tree is a path of length with one of its end vertices being the root and all the elements of mapped to the other end vertex. The following lemma gives us a way to modify a depthdecomposition.
Lemma 3.2.
Let be a finite matroid. If is a depthdecomposition of , then there is a depthdecomposition such that is a leaf of for every element of .
Proof.
Let be a depthdecomposition of . For every inner vertex of , let be a leaf of that is a descendant of . For every , define as follows.
We now verify that is a depthdecomposition of . The part (1) of Definition 3.1 holds since we have not changed the tree . To check part (2), observe that for any subset of the elements of , the subtree with respect to is contained in the subtree with respect to . Hence, is a depthdecomposition of . ∎
As the notion of graph treedepth [32], the parameter of matroid branchdepth is also minor monotone.
Proposition 3.3.
If is a minor of , then .
Proof.
Since a minor of a matroid is obtained by a sequence of contractions and deletions of some of its elements, it is enough to show that if is a matroid and is an element of , then the branchdepth of both and is at most . Fix a matroid and . Let be a depthdecomposition of of depth . By Lemma 3.2, we can assume that is a leaf of for every .
If is a loop in then . It is easy to see that for every we have . Hence, is a depthdecomposition of .
We now assume that is not a loop. Let , let be the leaf , and let be the parent of . Set and define as follows:
We now show that is a depthdecomposition of . Since is not a loop, we have . Thus, . Now, consider a subset . Recall that . If we employ the bound on the rank function provided by the depthdecomposition of :
Otherwise, we have
Let . If is a bridge then . Hence, we may assume that is not a bridge in . In this case, we claim that is a depthdecomposition of . Since the rank of equals the rank of , we have , and it also holds that for every . ∎
If a graph has a path with vertices, its treedepth (see Definition 3.5 if needed) is at least , see e.g. [32, Chapter 6]. The next proposition relates the length of circuits in a matroid to its branchdepth, in the analogy to the relation between the graph treedepth and the existence of long paths.
Proposition 3.4.
Let be a matroid and the size of its largest circuit. Then .
Proof.
Let be the matroid that consists of exactly one circuit of size . If has a circuit of length , then contains as a minor. Hence, it is enough to show by Proposition 3.3 that the branchdepth of is at least . We prove this statement by induction on .
Let be a depthdecomposition of such that has depth and such that is a leaf of for every . Its existence follows from Lemma 3.2.
Let be the root of . We first prove that the degree of is one. Suppose not. Let be vertices of one of the subtrees of , and let be the subtree induced by and the subtree induced by the vertices not contained in . Observe that for . It follows that , which is impossible.
Let be a vertex of of degree larger than two that is as close to the root as possible. If there is no such vertex, is a path and it has depth .
Let be a path from to , its length, and vertices of one of the subtrees of . Let be the subtree induced by and the subtree induced by and the vertices not contained in or in . Further, let and for . Observe that both and are nonzero, and that
(1) 
Since is a proper subset of , it is independent and we get that for . This yields that ; otherwise, by (1). By symmetry, we may assume that , which gives .
Let . Observe that is isomorphic to . Let be the tree obtained by considering a path of length of , identifying one of its end vertices with the root of the tree and rooting the resulting tree at the other end vertex of the path. Observe that since the number of its edges is smaller by compared to , and that the tree with is a depthdecomposition of . By induction, the depth of is at least . Since , it follows that the depth of is at least . ∎
We now recall the notion of graph treedepth. We use to denote the transitive closure of a rooted tree , i.e., the graph with vertex set and an edge connecting each pair of vertices and such that is an ancestor of in .
Definition 3.5.
The treedepth of a graph is the smallest possible depth of a rooted tree with the same vertex set as such that . Such a tree is called an optimal treedepth decomposition of .
We next relate the branchdepth of a graphic matroid to the treedepth of the underlying graph.
Proposition 3.6.
The branchdepth of a graphic matroid is at most .
Proof.
Let be a graph on vertices and let be the corresponding graphic matroid. We proceed by induction on . If or , the claim holds. If is not 2connected, let be its 2connected components (blocks). Since the treedepth is a minor monotone parameter, each has treedepth at most and the matroid has a depthdecomposition with depth at most by induction. Since the matroid is the disjoint union of the matroids , a depthdecomposition of can be obtained by identifying the roots of depthdecompositions of . The depth of such a depthdecomposition is at most and the claim follows. So, we assume that is 2connected in the rest of the proof.
Let be an optimal treedepth decomposition of . We construct a depthdecomposition of as follows. The function maps an element to the end vertex of that is farther from the root of . We verify the two conditions from Definition 3.1. Since is connected, we indeed have as required by the condition (1). Consider a subset . We show that to establish the condition (2). We may assume that has no circuit: if contained a circuit, removing an element from a circuit of would not change and it could not increase . So, we assume that is independent and .
Let be the edge sets of the connected components of the graph , and let be the their vertex sets. Further, let be the set . Note that the sets are disjoint and is a subset of . Since is connected, the set contains a unique vertex closest to the root, and is equal to with the vertex closest to the root removed. Since the subtree contains an edge from each vertex of to its parent, and the sets are disjoint (and so are the sets ), is at least . Since the rank of is equal to the sum of ranks of the sets and , it follows that . This finishes the proof of the lemma. ∎
Note that the converse of Proposition 3.6 does not hold. The graphic matroids of an vertex star and an vertex path are isomorphic and both have branchdepth one despite of the treedepth of being one and the treedepth of being . Nevertheless, the following inequality holds for 2connected graphs.
Proposition 3.7.
Let be a 2connected graph with treedepth . Then, the branchdepth of a graphic matroid is at least .
3.2 Technical lemmas
In this section, we establish further properties of the branchdepth, which are important to prove the correctness of the algorithm presented later. The following two claims follow directly from the definition of contracting an element of a matroid.
Lemma 3.8.
Let be a circuit in a matroid . Let . If , then the set is a circuit in .
Proof.
By the definition of contracting an element, it follows that . On the other hand, if is a proper subset of , then . Hence, is a circuit in . ∎
Lemma 3.9.
Let be a matroid and an element of . If is a circuit of , then has a circuit such that .
Proof.
First observe that any proper subset of is independent in : indeed, if is a proper subset of , then . If is not independent in , then is a circuit.
Suppose that is independent in . We claim that is a circuit. First, , i.e., is not independent. Let be a subset of . We have already observed that is independent if . If , then , i.e., is independent. We conclude that is a circuit. ∎
When encountering a circuit, the algorithm is going to proceed by contracting one of its elements. The following lemma will be crucial for the analysis.
Lemma 3.10.
Let be a connected matroid, an element of such that is disconnected, and let be the components of . For every circuit of containing , there exists such that .
Proof.
Suppose that contains an element from and an element from , . Let be the union of , and let and . Hence, we get the following
Since are the components of , it follows that . However, is a circuit in by Lemma 3.8. Since and are proper subsets of , both and are independent in . Hence, and are independent in and , respectively. It follows that , which is impossible. ∎
Lemma 3.11.
Let be a connected matroid. Let be an element of such that is not connected and let be the components of . For each there is a circuit in containing such that .
Proof.
Fix . Since is connected, there is a circuit containing any two elements of , in particular, has a circuit containing the element and an element of . By Lemma 3.10, the circuit must be a subset of . ∎
The following lemma allows us to find an obstruction to a small branchdepth. We utilize this lemma to show that Algorithm 3.3 always returns a depthdecomposition of depth at most . Figure 1 contains an illustration of the notation used in the lemma.
Lemma 3.12.
Let be a matroid. Let be distinct elements of and subsets of such that
Let and , . Further, set
If is a circuit in for , then contains a circuit of length at least containing .
Proof.
We prove the statement by induction on . For it suffices to take the circuit itself.
Let . By induction, has a circuit of length at least that contains . Let . Since is a circuit, is independent in and thus in . Also note .
Let . Since is a circuit in , is a circuit in by Lemma 3.8. Furthermore, it holds that . If is a circuit in , then there is a circuit in by Lemma 3.9. Therefore, it suffices to find a circuit of length at least in .
We will show that or is a circuit in . Since is independent in , we get that is independent in . We next show that
(2) 
for any , and for any set . This follows from the following application of the submodularity of the rank function:
Hence, for any proper subset , we have
where the last equality follows from the fact that is a circuit in . Thus, both and are independent in . On the other hand, it also holds that
where the first equality follows from (2), the second from , and the last from the fact that is a circuit in . Consequently, neither and is independent in . To finish the proof, it suffices to show that or is independent in . This can be shown using the submodularity of the rank function and (2) as follows: