 I'm working at the university of Hamburg on infinite matroid theory.

# A breakthrough on the matroid intersection conjecture

Welcome to the next post in the series on infinite matroids, where I am excited to report some recent progress by Attila Joó on the matroid intersection conjecture. Before that, let’s take some time to get familiar with the conjecture itself. It pops up all over the place in infinite matroid theory. More precisely, there are a number of central theorems of finite matroid theory stating that certain natural bounds on the sizes of sets are tight. In generalising these to infinite matroids, it is often helpful to focus on the combinatorial structures which witness the tightness of the bound. The statement that such structures exist in the infinite case almost always turns out to be a facet of the matroid intersection conjecture. A full explanation would be too long to fit in this post, but luckily it was the subject of the previous post in this series, so please look there if you are interested in more details.

So what does the conjecture say? Let’s say that a pair $(M,N)$ of matroids on the same ground set $E$ has the intersection property if there are a partition of $E$ as $P \cup Q$ and a subset $I$ of $E$, independent in both $M$ and $N$, such that $I \cap P$ is spanning in $M$ and $I \cap Q$ is spanning in $N$. This is exactly the kind of combinatorial structure which would witness the tightness of the bound in the matroid intersection theorem if $M$ and $N$ were both finite. The matroid intersection conjecture states that any such pair $(M,N)$ has the intersection property.

We saw in the previous post that there is a close link between this conjecture and packings and coverings of families of matroids. Recall that a base packing of a family $(M_k)_{k \in K}$ of matroids on the same ground set $E$ consists of a family $(S_k)_{k \in K}$ of bases in the respective matroids, all disjoint from each other, whereas a base covering of such a family consists of a family $(I_k)_{k \in K}$ of bases in the respective matroids whose union is the whole ground set $E$. Finally, we say that such a family of matroids has the packing/covering property if we can partition $E$ as $P \dot \cup Q$ into a packable part $P$ and a coverable part $Q$, such that $(M_k | P)_{k \in K}$ has a base packing and $(M_k.Q)_{k \in K}$ has a base covering.

The relation of this to intersection is that $(M,N)$ has the intersection property if and only if $(M,N^*)$ has the packing/covering property. Thus by simply dualising one of the matroids we see that it follows from the matroid intersection conjecture that any pair of matroids has the packing/covering property. The packing/covering conjecture is that any family at all of matroids has the packing/covering property.

Although this seems like a far-reaching generalisation, in fact it is already encompassed by the original intersection conjecture. To see why, we have to closely examine a construction which allows us to encode packings and coverings for a family $(M_k)_{k \in K}$ of matroids in terms of packings and coverings for a carefully chosen pair of matroids, namely the direct sum $M := \bigoplus_{k \in K} M_k$ of all the matroids in the family and a direct sum $N := \bigoplus_{e \in E} U_{1,K}^*$ of circuits of size $|K|$, one for each element of the ground set $E$.

Note that both these matroids have the same ground set $K \times E$. We can think of the elements of $K \times E$ as being laid out in a big grid, with the rows indexed by elements of $E$ and the columns by elements of $K$. So $M$ is obtained by putting a copy of $M_k$ in the $k$th column for each $k$, whereas $N$ is obtained by making each row into a circuit.

What do packings for this pair of matroids look like? Such a packing must consist of bases $S_1$ and $S_2$ in the respective matroids. Let’s think about $S_2$ first, since the structure of $N$ is a little simpler. It must consist of a basis of each row, and since each row is just a circuit, $S_2$ must contain all but precisely one element of each row. On the other hand, $S_1$ consists of a base $S_k$ in each column of the corresponding matroid $M_k$. Since $S_1$ and $S_2$ have to be disjoint, $S_1$ can contain at most one element from each row. That precisely means that the $S_k$ form a base packing for the $M_k$.

The story for coverings is similar. Any covering for $(M,N)$ must consist of bases $I_1$ and $I_2$ in the respective matroids. Once more $I_2$ must contain all but precisely one element of each row. Since the union of $I_1$ and $I_2$ must be the whole ground set, this enforces that $I_1$ must have at least one element in each row, meaning that the bases $I_k$ form a base covering for the $M_k$.

Combining these ideas, we can see that everything works in the same way for the packing/covering property as well. First of all, if we have a partition of $E$ as $P \dot \cup Q$ witnessing the packing/covering property for the $M_k$ then the arguments above show that $\hat P := K \times P$ and $\hat Q = K \times Q$ give a partition of $K \times E$ witnessing the packing/covering property there too.

The opposite direction seems a little trickier; given a partition of $K \times E$ into sets $\hat P$ and $\hat Q$ witnessing the packing/covering property for $M$ and $N$, there is no reason why $\hat P$ and $\hat Q$ should have the form $K \times P$ and $K \times Q$ for a partition of $P$ and $Q$: there could easily be some row $R$ which meets both $\hat P$ and $\hat Q$. But note that for any such row, the restriction of $N$ to $R \cap \hat P$ is independent, and so in the packing $(S_1, S_2)$ of $(M | \hat P, N | \hat P)$ we have $R \cap \hat P \subseteq S_2$, and so $S_1$ is disjoint from $R$.

This suggests that we should shift all such rows completely into the $Q$-side of the partition. More formally, we let $P$ be the set of all $e \in E$ whose corresponding row is completely included in $\hat P$. By the above argument we get a packing of $M | (K \times P)$ and $N | (K \times P)$, and so a packing of $(M_k | P)_{k \in K}$. Let $Q$ be the complement of $P$ in $E$: the set of all $e \in E$ whose corresponding row meets $\hat Q$. In the covering $(I_1, I_2)$ of $(M.\hat Q, N.\hat Q)$, it is still true that $I_2$ is missing at least one element from each row meeting $\hat Q$ and so we also get a covering of $(M_k.Q)_{k \in K}$ as above.

In summary, we see that the family $(M_k)_{k \in K}$ has the packing/covering property if and only if the pair $(M,N)$ does. Hence the packing/covering conjecture for arbitrary families of matroids is equivalent to the packing/covering conjecture for pairs of matroids. Both of these statments, as well as the matroid intersection conjecture, can be seen as different points of view on the same problem. The reductions sketched so far were worked out a while ago by Johannes Carmesin and myself in .

Now let’s turn our attention to the progress which has been made so far towards proving the intersection and packing/covering conjectures. More precisely, let’s consider for which special cases the intersection property or the packing/covering property have been proved to hold. Since almost nothing is known in the case that the matroids are uncountable, in the following discussion we will only consider countable matroids. So from now on you can assume that the common ground set of any family of matroids we consider is countable.

There are two main types of matroid for which the packing/covering conjecture is known. The first deals with matroids built up by gluing together a tree of finite matroids, according to a recipe outlined in a previous post in this series. Together with Johannes Carmesin, I was able to show  that any two matroids built according to this recipe along the same tree structure will have the packing/covering property. Since duality is preserved by this construction, and $(M,N)$ has the intersection property precisely when $(M,N^*)$ has the packing/covering property, it follows that any two such matroids also have the intersection property.

The second is the situation in which the matroids are close to being finitary or cofinitary. The first progress in this direction was by Aharoni and Ziv , who showed that $M$ and $N$ have the matroid intersection property whenever $M$ is finitary and $N$ is a countable direct sum of matroids of finite rank. Translating in the usual way by dualising $N$, we see that $M$ and $N$ have the packing/covering property whenever $M$ is finitary and $N$ is a countable direct sum of matroids of finite co-rank.

Using the ideas above, we can derive a much more general result from this one. Suppose that we have any family $(M_k)_{k \in K}$ of finitary matroids, from which we build matroids $M := \bigoplus_{k \in K} M_k$ and $N := \bigoplus_{e \in E} U_{1,K}^*$ as above. Then $M$ will still be finitary, and by definition $N$ will be a sum of a countable family of circuits, which have finite co-rank. By Aharoni and Ziv’s result, $M$ and $N$ has the packing/covering property, and hence so does the family $(M_k)_{k \in K}$. So any family of finitary matroids has the packing/covering property!

We can use similar tricks to obtain the same result for families of cofinitary matroids. We begin again with Aharoni and Ziv’s result, but this time we begin by dualising the other matroid. This tells us that $M$ and $N$ have the packing/covering property whenever $M$ is cofinitary and $N$ is a countable direct sum of matroids of finite rank. If we set $M := \bigoplus_{k \in K} M_k$ and $N := \bigoplus_{e \in E} U_{1,K}^*$ as above, with $(M_k)_{k \in K}$ now a family of cofinitary matroids, then $M$ will certainly be cofinitary, but the circuits $U_{1,K}^*$ will only be of finite rank if $K$ is finite. So it seems that we only obtain that any finite family of cofinitary matroids has the packing/covering property.

But we can press on further. In particular, we now know that any pair $(M,N)$ of cofinitary matroids will have the packing/covering property. If $M$ and $N$ are given as usual by $\bigoplus_{k \in K} M_k$ and $\bigoplus_{e \in E} U_{1,K}^*$, where $(M_k)_{k \in K}$ is an arbitrary family of cofinitary matroids, then both $M$ and $N$ will be cofinitary, so this pair will have the packing/covering property. Thus by bootstrapping in this way we obtain that any family of cofinitary matroids has the packing/covering property.

Using the fact that $(M,N)$ has the intersection property precisely when $(M,N^*)$ has the packing/covering property, either of these two results can be used to obtain the unsatisyfingly asymmetrical statement that any pair consisting of a finitary and a cofinitary matroid has the intersection property.

This was essentially the state of the art until recently, although together with Johannes Carmesin, Shadisadat Ghaderi and Jerzy Wojciechowski I had developed a technique  to extend these results slightly: to matroids which are not quite finitary or cofinitary, but are very close. Thus any family of so-called nearly finitary matroids and any family of nearly cofinitary matroids has the packing/covering property. Any pair consisting of a nearly finitary and a nearly cofinitary matroid has the intersection property.

Recently, however, there has been a remarkable breakthrough: Let’s say that a matroid is simple if all of its components are finitary or cofinitary. Attila Joó has shown  that any pair of (countable) simple matroids has the intersection property. Although this may seem similar to the results listed above, it is considerably stronger.

First of all, dualising in the usual way, we can see that it implies that if any pair of simple matroids also has the packing/covering property. Building $M$ and $N$ in the usual way from a family of simple matroids, it follows that any family of simple matroids has the packing/covering property.

In particular, Attila’s result implies the results from earlier that any family of finitary matroids or of cofinitary matroids has the packing/covering property. But its true depth is made clear from the fact that (still in the countable setting) it implies Aharoni and Berger’s deep and far-reaching generalisation of Menger’s theorem for undirected graphs, as explained in the previous post in this series.

There are still a number of questions in this area where we can hope for future progress. Most pressing is the question of whether Attila’s result can be extended to uncountable matroids. More ambitiously, there remains the question of whether the restrictions of finitariness or cofinitariness can be substantially loosened, or even entirely dropped.

 Bowler, N., & Carmesin, J. (2015). Matroid intersection, base packing and base covering for infinite matroids. Combinatorica, 35, 153-180.
 Bowler, N. & Carmesin, J. (2021+). On the intersection conjecture for infinite trees of matroids, preprint available at https://arxiv.org/abs/1404.6067
 Aharoni, R., & Ziv, R. (1998). The intersection of two infinite matroids. Journal of the London Mathematical Society, 58(3), 513-525.
 Bowler N, Carmesin J, Wojciechowski J, Ghaderi S. (2020). The Almost Intersection Property for Pairs of Matroids on Common Groundset. The Electronic Journal of Combinatorics. 2020,Jul 10:P3-5.
 Joó, A. (2021+).On the packing/covering conjecture of infinite matroids, preprint available at https://arxiv.org/abs/2103.14881

# The infinite matroid intersection conjecture

Today we’ll return to our examination of infinite matroids. So far we saw why they are defined the way they are and what the known examples look like. Then we examined a very flexible way of building infinite matroids from trees of finite matroids and saw how to use that construction as a tool in topological infinite graph theory.

The aim today is to understand the deepest and most important unproved conjecture about infinite matroids, the infinite matroid intersection conjecture. We won’t be looking at progress towards the conjecture today, just approaching the statement from a number of angles and getting a sense of its connections to various very different-looking problems in infinite combinatorics. I hope that by the end of the post you will be convinced, as I am, that it is a deep and compelling problem. Here it is:

Conjecture (Nash-Williams): Let $M$ and $N$ be (possibly infinite) matroids on the same ground set $E$. Then there are a subset $I$ of $E$ and a partition of $E$ into sets $P$ and $Q$ such that $I$ is independent in both $M$ and $N$, $I \cap P$ spans $P$ in $M$ and $I \cap Q$ spans $Q$ in $N$.

Like a TARDIS, at a first glance this statement seems simple and perhaps a little odd, and its deeper significance is hidden. To get a sense of that significance, we must go on a long journey and see how it materialises within apparently widely separated worlds.

Our journey begins with the observation that finding good infinite versions of theorems about finite combinatorial objects is hard. All too often, the obvious generalisation is either straightforwardly false or else is a simple consequence of the finite version of the theorem, and as such has no new content.

An example of the latter phenomenon is Menger’s Theorem. If $G$ is a graph and $A$ and $B$ are sets then an $A$-$B$ separator in $G$ is defined to be a set $S$ of vertices of $G$ such that there is no path from $A$ to $B$ in $G – S$. Menger’s theorem states that if $G$ is finite then the minimal size of an $A$-$B$ separator in $G$ is the same as the maximal size of a set of disjoint paths from $A$ to $B$ in $G$.

The obvious way to generalise this statement to infinite graphs would be to simply replace the word ‘size’ with the word ‘cardinality’ in both places where it appears. However, the statement obtained in this way has no more content than the finite version of the theorem. We can see this by considering an $A$-$B$ separator $S$ of minimal cardinality.

If $S$ is infinite, then any set of fewer than $|S|$ paths from $A$ to $B$ uses fewer than $|S|$ vertices, and so cannot be maximal. So in that case the statement is clear, and we can suppose instead that $|S|$ is some natural number $n$. Now for each $m \leq n$ we can easily build a finite subgraph $G_m$ of $G$ in which any $A$-$B$ separator has size at least $m$: we may take $G_0$ to be empty, and build $G_{m+1}$ from $G_m$ by adding a path $P_X$ of $G$ from $A$ to $B$ avoiding each set $X$ of $m$ vertices of $G_m$. Then by Menger’s theorem $G_n$ already contains $n$ disjoint paths from $A$ to $B$.

It was Paul Erdős who saw how to get a much deeper infinite generalisation by first reformulating Menger’s theorem as a structural statement. Suppose that we consider an $A$-$B$ separator $S$ of minimal size and a set $\cal P$ of disjoint paths from $A$ to $B$ of maximal size. Then each path in $\cal P$ contains at least one vertex in $S$, and these vertices must all be distinct since the paths are disjoint. But by Menger’s theorem there can only be as may paths in $\cal P$ as there are vertices in $S$. So $S$ must consist of one vertex on each path in $\cal P$.

So it follows from Menger’s theorem that in a finite graph $G$ we can always find a set $\cal P$ of disjoint $A$-$B$ paths together with an $A$-$B$ separator $S$ consisting of one vertex from each path in $\cal P$. On the other hand, this structural statement also implies Menger’s theorem. After all, if ${\cal P}’$ is a set of disjoint paths from $A$ to $B$ of maximal size and $S’$ is an $A$-$B$ separator of minimal size then $|S’| \leq |S| = |{\cal P}| \leq |{\cal P}’|$. But also $|{\cal P}’| \leq |S’|$ since each path in ${\cal P}’$ must contain a different point of $S’$. So $|{\cal P}’| = |S’|$, as desired.

Erdős’ generalisation of Menger’s theorem is therefore the following structural statement:

Theorem (Aharoni and Berger): Let $G$ be a (possibly infinite) graph and let $A$ and $B$ be sets. Then there is a set ${\cal P}$ of disjoint $A$-$B$ paths together with an $A$-$B$ separator $S$ consisting of one vertex from each path in ${\cal P}$.

This statement contains some serious content about the structure of infinite graphs, and it remained open for almost half a century before finally being proved by Aharoni and Berger in 2009 [AB09]. Their proof remains one of the deepest ever given in infinite combinatorics.

Another example of the difficulties of generalisation from finite to infinite objects is given by the tree packing and covering theorems. The tree covering theorem states that a connected graph $G$ is a union of $k$ spanning trees if and only if for any set $X$ of vertices of $G$ the induced subgraph $G[X]$ has at most $k(|X| – 1)$ edges, and the tree packing theorem states that a connected graph $G$ includes $k$ edge-disjoint spanning trees if and only if for any partition $P$ of the vertex set of $G$, the quotient graph $G/P$ has at least $k(|P|-1)$ edges. Here $G/P$ is the graph whose vertices are the partition classes and whose edges are those of $G$ which go between partition classes, with endpoints the partition classes which they join.

Once more, the obvious generalisation of the tree covering theorem to infinite graphs has no more content than the finite version of the theorem; it can be proved from it by a straightforward compactness argument. On the other hand the obvious generalisation of the tree packing theorem to infinite graphs is false; there is a counterexample due to Aharoni and Thomassen [AT89]. And once more, to find the correct infinite version of the theorems we must begin by finding a structural version in the finite case. Indeed, it turns out that the tree packing and covering theorems have a unifying structural generalisation:

Theorem ([D17, Theorem 2.4.4]): Let $G$ be a connected finite graph and $k$ a natural number. Then there is a partition $P$ of the vertex set of $G$ such that $G/P$ is a union of $k$ spanning trees and $G[X]$ is connected and has $k$ edge-disjoint spanning trees for each partition class $X$ of $P$.

This tree packing/covering theorem implies both the tree packing theorem and the tree covering theorem. For tree packing, the necessity of the condition is clear, so it suffices to prove sufficiency. We can do this by applying the condition to the partition $P$ given by the tree packing/covering theorem. This gives that $G/P$ has at least $k(|P|-1)$ edges. Since it is a union of $k$ spanning trees, those trees must be edge disjoint. Combining these with the edge-disjoint spanning trees in each $G[X]$ gives $k$ edge-disjoint spanning trees in $G$. The derivation of the tree covering theorem from the packing/covering theorem is similar.

This gives us a nontrivial common generalisation of the tree packing and covering theorems to infinite graphs: we can simply omit the word ‘finite’ from the tree packing/covering theorem. The proof of this generalisation, though much simpler than that for the infinite version of Menger’s theorem, goes beyond the scope of this post.

We have now seen two examples where, to find the correct infinite generalisation of a theorem about finite graphs, it was necessary to first reformulate the finite theorem as a structural result. The same is true for theorems about finite matroids, but in this case something remarkable happens. The infinite structural statement you get is usually just the infinite matroid intersection conjecture!

This is not too surprising for the matroid intersection theorem, since Nash-Williams formulated the intersection conjecture to be an infinite structural generalisation of that statement. Recall that the matroid intersection theorem states that the largest size of a common independent set of two matroids $M$ and $N$ on the same ground set $E$ is the same as the minimum value over all partitions of $E$ into sets $P$ and $Q$ of $r_M(P) + r_N(Q)$. The inequality one way around is clear, since if $I$ is independent in both $M$ and $N$ and $\{P, Q\}$ is a partition of $E$ then $|I| = |I \cap P| + |I \cap Q| \leq r_M(P) + r_N(Q)$. For this inequality to be an equality, we must have that $I \cap P$ spans $P$ in $M$ and $I \cap Q$ spans $Q$ in $N$, just as in the conjecture.

There are some less expected cases. Let’s consider Tutte’s linking theorem, the closest matroidal analogue of Menger’s theorem. Let $M$ be a finite matroid with ground set $E$, and let $A$ and $B$ be disjoint subsets of $E$. Let $E’ := E \setminus (A \cup B)$. Then the connectivity $\lambda_M(A, B)$ from $A$ to $B$ in $M$ is defined to be the minimal value of $\kappa_M(A \cup P)$ over all bipartitions of $E’$ into sets $P$ and $Q$. Here $\kappa_M$ is the connectivity function of $M$, given by $\kappa_M(X) := r_M(X) + r_M(E \setminus X) – r(M)$. The linking theorem states that the maximum value of $\kappa_N(A)$ over all minors $N$ of $M$ with ground set $A \cup B$ is $\lambda_M(A,B)$.

It turns out that there is a structural analogue of this statement. Each such minor $N$ must have the form $M/I\backslash J$, where $I$ and $J$ form a partition of $E’$. By moving loops of $M/I$ into $J$ if necessary, we may suppose that $I$ is independent. We may now calculate as follows:

$\kappa_{M/I \setminus J}(A) = (r(A \cup I) – |I|) + (r(B \cup I) – |I|) – (r(M) – |I|) \\= (r(A \cup I) – |Q \cap I|) + (r(B \cup I) – |P \cap I|) – r(M) \\ \leq r(A \cup (I \cap P)) + r(B \cup (I \cap Q)) – r(M) \\ \leq r(A \cup P) + r(B \cup Q) – r(M) \\ = \kappa_M(A \cup P)$

So equality of the left and right sides is equivalent to the statement that each inequality in the above calculation is an equality, giving the following four conditions:

1. $I \cap P$ spans $P$ in $M/A$
2. $I \cap Q$ spans $Q$ in $M/B$
3. $I \cap P$ is independent in $M/(B \cup (I \cap Q))$
4. $I \cap Q$ is independent in $M/(A \cup (I \cap P))$

The outlines of our TARDIS are beginning to materialise. Indeed, consider a minimal set $I$ satisfying these conditions. By minimality, $I \cap P$ will be independent in $M/A$ and $I \cap Q$ will be independent in $M/B$. Thus $I$ itself will be independent in both matroids. To put it another way, $I$, $P$ and $Q$ will witness that $M/A \backslash B$ and $M \backslash A/B$ satisfy the matroid intersection conjecture.

Thus the infinite generalisation of Tutte’s linking theorem is the statement that, for any (possibly infinite) matroid $M$ and any disjoint sets $A$ and $B$ of elements of $M$, the matroids $M/A\backslash B$ and $M \backslash A/B$ satisfy the infinite matroid intersection conjecture. Given this connection, it should not be too surprising that Aharoni and Berger’s infinite generalisation of Menger’s theorem follows from the infinite matroid intersection conjecture. Precise details of the derivation can be found in [ACF18].

What about the tree packing and covering theorems? Their matroidal analogues are the base packing and covering theorems, which in their full generality apply to a list $M_1, M_2, \ldots M_k$ of finite matroids on a common ground set $E$. A base packing for such a list is a collection of disjoint bases, one from each $M_i$. A base covering for such a list is a collection of bases, one from each $M_i$, whose union is the whole of $E$. The base packing theorem states that there is a base packing precisely when for any subset $Q$ of $E$ we have $\sum_{i = 1}^k r(M_i.Q) \leq |Q|$, and the base covering theorem states that there is a base covering precisely when for any subset $P$ of $E$ we have $\sum_{i = 1}^k r(M_i | P) \geq |P|$.

Once more we can combine these statements into a unified structural statement, the base packing/covering theorem, which states that given such a list of finite matroids on $E$ we can find a bipartition of $E$ into sets $P$ and $Q$ such that the matroids $M_1 | P, \ldots M_k | P$ have a packing and the matroids $M_1.Q, \ldots M_k.Q$ have a covering. The derivations of the base packing and covering theorems from this statement are analogous to the derivation of the tree packing theorem from the tree packing/covering theorem above. So the infinite version of the base packing and covering theorems is given by the same statement applied to a family of infinite matroids. We shall call this the base packing/covering conjecture.

Let’s consider the special case $k = 2$ in more detail. The existence of a packing for $M_1 | P$ and $M_2 | P$ is equivalent to the existence of a subset $I_P$ of $P$ such that $I_P$ spans $P$ in $M_1$ and $P \setminus I_P$ spans $P$ in $M_2$. Similarly the existence of a covering for $M_1.Q$ and $M_2.Q$ is equivalent to the existence of a subset $I_Q$ of $Q$ such that $I_Q$ is independent in $M_1/P$ and $Q \setminus I_Q$ is independent in $M_2/P$. Since a set is independent in a matroid precisely when its complement is spanning in the dual matroid, we can rephrase these conditions as follows:

1. $I_P$ spans $P$ in $M_1$
2. $I_Q$ spans $Q$ in $M_2^*$
3. $I_P$ is independent in $M_2^*/Q$
4. $I_Q$ is independent in $M_1/P$

Once again, as if from nowhere, the TARDIS appears. If we choose $I_P$ and $I_Q$ minimal subject to conditions (i) and (ii) then they will still satisfy conditions (iii) and (iv), which will guarantee that $I:=I_P \cup I_Q$ is independent in both $M_1$ and $M_2^*$, meaning that $I$, $P$ and $Q$ witness that $M_1$ and $M_2^*$ satisfy the infinite matroid intersection conjecture.

The TARDIS not only appears in unexpected places, it is also bigger on the inside than it seems. For example, the remarks in the last couple of paragraphs only apply to pairs of matroids, that is, to lists of length 2. But in fact it is possible to derive the full base packing/covering conjecture from the special case of pairs, and hence from the infinite matroid intersection conjecture. We will see the reasons for this when we look at the structure of the conjecture more closely in the next post in the series. For now we just note the consequence that the tree packing/covering theorem mentioned earlier also follows from the infinite matroid intersection conjecture.

We have seen how the infinite matroid intersection conjecture arises naturally as the infinite structural analogue of the matroid intersection theorem, the linking theorem, and the base packing and covering theorems. The same also holds for the matroid union theorem, which we do not have space to discuss here [BC15]. Thus the process of finding an infinite generalisation of all these statements reveals their unified structural heart. In the next post we will examine that structural heart more closely, looking at just what sort of structure the conjecture gives us, and we will survey the special cases for which the conjecture is already known.

Bibliography:

[AB09] R. Aharoni and E. Berger, Menger’s Theorem for Infinite Graphs, Inventiones mathematicae 176(1):1–62 (2009).

[ACF18] E. Aigner-Horev, J. Carmesin and J.-O. Fröhlich, On the Intersection of Infinite Matroids, Discrete Mathematics 341(6):1582-1596 (2018).

[AT89] R. Aharoni and C. Thomassen, Infinite, highly connected digraphs with no two arc-disjoint spanning trees. J. Graph Theory, 13:71–74 (1989).

[BC15] N. Bowler and J. Carmesin, Matroid Intersection, Base Packing and Base Covering for Infinite Matroids, Combinatorica 35(2):153-180 (2015).

[D17] R. Diestel, Graph Theory, 5th edition, Springer-Verlag (2017).

# The matroid union is back!

It has been almost a year, but now the Matroid Union is back. Just as before, we’re planning to have a new post every 2 weeks on Monday. Over the next month Laura Anderson will be writing about a new application of oriented matroids in mathematical psychology, and I will introduce the deepest problem in the theory of infinite matroids, the infinite matroid intersection conjecture.

The team of core contributors has changed a little; we’re sorry to say goodbye to Stefan van Zwam and Irene Pivotto, who put a lot of energy into making this blog what it is. But we have some fresh new contributors: Laura Anderson and Nick Brettell. You can get some idea of what topics we are each hoping to cover here. A variety of other topics will be covered by guest posts. If you have any ideas for topics which you would like to see on the blog, or even which you would like to write about yourself, then please get in touch.

# Building matroids from infinite graphs

Today we’ll be looking at infinite matroids again. We started this series by examining the question of how infinite matroids could be defined. With a suitable definition in hand, we took a quick tour through the zoo of known examples. Then we took a closer look at one very flexible way to build infinite matroids: by sticking together infinite trees of matroids.

In that construction we used 2-sums to stick the matroids together. Suppose that we have a finite collection of matroids arranged in a tree, so that their ground sets overlap (always in single edges) if and only if they are adjacent in the tree. Then because 2-sums are associative we have an unambiguously defined 2-sum of that collection. In the previous post in this series, we saw that this construction also works if we allow the tree to be infinite, but that we have to specify a little extra information the set $\Psi$ of ends of the tree which the circuits are allowed to use.

The same trick can be used for other kinds of associative sum of finite matroids. In this post we’ll see how it works for sums of representable matroids, and why that is useful for understanding the topological spaces associated to infinite graphs.

To see how the addition of representable matroids works, we need to look at them from a slightly unusual angle. Let’s say that we have a matroid $M$ on a ground set $E$, and suppose that we have vectors $\{v_e | e \in E\}$ in some vector space over a field $k$, giving us a representation of $M$ over $k$. Then we can capture all the essential information about this representation by forgetting the details of the vector space and focusing just on the linear relationships amongst the vectors. More formally, we say that an element $\lambda$ of the space $k^E$ is a linear dependence of the $v_e$ if $\sum_{e \in E} \lambda(e)v_e = 0$. Then the linear dependences form a subspace $V$ of $k^E$, and this subspace is enough to recover the matroid $M$; the circuits of $M$ are precisely the minimal nonempty supports of vectors in $V$. For those who prefer to think of representations in terms of matrices rather than families of vectors, the subspace we’re working with is just the orthogonal complement of the row space of the matrix.

So we can encode representations of matroids on $E$ over $k$ as subspaces of $k^E$. This way of seeing representations fits well with matroid duality, in that if $V \subseteq k^E$ represents a matroid $M$ on $E$ then the orthogonal complement $V^{\bot}$ represents the dual matroid $M^*$. If we define $M(V)$ to be the matroid whose circuits are the minimal nonempty supports of elements of $V$, then we can express this as $M(V^{\bot}) = (M(V))^*$.

The advantage of this perspective is that there is a natural way to glue together such subspaces, which we can use to build a self-dual gluing operation for represented matroids. Suppose that we have two sets $E_1$ and $E_2$ and subspaces $V_1$ and $V_2$ of $k^{E_1}$ and $k^{E_2}$, respectively. We now want to glue these together to give a subspace $V_1 \oplus V_2$ of $E_1 \triangle E_2$. As with the 2-sum, we throw away the gluing edges’ in the overlap $E_1 \cap E_2$. The idea is to take pairs of vectors which match on the gluing edges, patch them together and throw away the part supported on the gluing edges. More precisely, we set $V_1 \oplus V_2 := \{v \mathord{\upharpoonright}_{E_1 \triangle E_2} | v \in k^{E_1 \cup E_2}, v \mathord{\upharpoonright}_{E_i} \in V_i\}$.

Like the 2-sum, this definition is self-dual in the sense that $(M(V_1 \oplus V_2))^* = M(V_1^{\bot} \oplus V_2^{\bot})$. It is also associative, in that if $V_1$, $V_2$ and $V_3$ are subspaces of $k^{E_1}$, $k^{E_2}$ and $k^{E_3}$ respectively and the sets $E_1$ and $E_3$ are disjoint then $(V_1 \oplus V_2) \oplus V_3 = V_1 \oplus (V_2 \oplus V_3)$. So if we have a finite collection of such representations on ground sets $E_t$ arranged in a finite tree, such that the ground sets only overlap if they are adjacent in the tree, then we have an unambiguous sum of all these subspaces.

Just as for 2-sums, we can also glue together infinite trees of represented matroids in this way, as long as we are careful to specify which ends of the tree the circuits are allowed to use. Formally, we do this as follows. Suppose that we have a tree $T$, a family of sets $E_t$ indexed by the nodes of $T$, such that $E_s \cap E_t$ is only nonempty if $s = t$ or $s$ and $t$ are adjacent in $T$, a family of subspaces $V_t \subseteq k^{E_t}$ and a Borel subset $\Psi$ of the set $\Omega(T)$ of ends of $T$. Then we can build a subspace of $k^{\bigtriangleup_{t}E_t}$ by setting

$\bigoplus^{\Psi}_t V_t := \{v \mathord{\upharpoonright}_{\bigtriangleup_t E_t} | v \in k^{\bigcup_t E_t}, v \mathord{\upharpoonright}_{E_t} \in V_t \text{ and } \Omega(T) \cap \overline{\{t | v \upharpoonright_{E_t} = 0\}} \subseteq \Psi\}$

and $M(\bigoplus^{\Psi}_t V_t)$ will be an infinite matroid.

What are these infinite sums good for? Well, if we have a representable matroid and we have a $k$-separation of that matroid then we can split it up as a sum of two matroids in this way such that there are fewer than $k$ gluing edges. We can use this to break problems about bigger matroids down into problems about smaller matroids. Similarly, if we have a nested collection of finite separations in an infinite matroid, cutting the ground set up into a tree of finite parts, then we can cut the matroid up into a sum of finite matroids and analyse its properties in terms of their properties. This kind of chopping up and reconstruction can also be helpful to show that the infinite object is a matroid in the first place.

Let’s see how that might work for a more concrete problem. Suppose that we have a locally finite graph $G$. Then we can build a topological space $|G|$ from it by formally adding its ends as new points at infinity (see for example [D10]). These spaces and their subspaces are key elements of topological infinite graph theory, which was where this series of posts started.

At first, it was hoped that these subspaces would have the nice property that if they are connected then they are path connected. But Agelos Georgakopoulos eventually found a counterexample to this claim [G07]. However, the set of ends used by the counterexample he constructed was topologically horrible, so we might still hope that if we have a connected subspace $X$ of $|G|$ such that the set $\Psi$ of ends contained in $X$ is topologically nice, then $X$ will be path-connected. Well, if we take topologically nice’ to mean `Borel’, then the ideas above let us show that this is true.

We can do this by considering the matroid whose circuits are the edge sets of those topological circles in $|G|$ which only go through ends in $\Psi$. More precisely, we need to show that this really does give the circuit set of a matroid $M(G, \Psi)$. If we can do that, then we can argue as follows:

Let $P$ be the set of edges which, together with both endpoints, are completely included in $X$. Let $u$ and $v$ be vertices in $X$. Build a new graph $G + e$ with an extra edge $e$ joining $u$ to $v$. Then since $X$ is connected, there can be no cocircuit of $M(G + e, \Psi)$ which contains $e$ and is disjoint from $P$ (such a cocircuit would induce a topological separation of $X$ with $u$ and $v$ on opposite sides). So $e$ is not a coloop in the restriction of $M(G + e, \Psi)$ to $P \cup \{e\}$. Hence there is a circuit through $e$ in that matroid, and removing $e$ from the corresponding topological circle gives an arc from $u$ to $v$ through $X$ in $|G|$. So any two vertices are in the same path-component of $X$. Similar tricks show the same for ends and interior points of edges.

What this argument shows is that the connection between connectivity and path-connectivity is encoded in the statement that $M(G, \Psi)$ is a matroid. To prove that statement, we can build $M(G, \Psi)$ as the sum of an infinite tree of graphic matroids in the sense described above. First of all, since $G$ is locally finite, we can cut it up into an infinite tree of finite connected parts using disjoint finite separators. Then we define the torso corresponding to a part to consist of that part together with new edges forming complete graphs on each of the separators. This gives us an infinite tree of finite graphs, and the ends of the tree correspond precisely to the ends of $G$. Now we take the graphic matroids corresponding to those graphs, take the standard binary representations of those matroids, and glue them together along this tree, taking the ends in $\Psi$ to be allowed for circuits. And presto! We have build the matroid $M(G, \Psi)$.

The details of this argument are explained in [BC15].

I can’t resist mentioning that the matroids we’ve just built in a bottom-up way also have a top-down characterisation. Consider the class of matroids whose ground set is the set of edges of $G$, and in which every circuit is a topological circuit of $G$ and every cocircuit is a bond of $G$. Let’s call such matroids $G$-matroids.

For some graphs $G$, we can find $G$-matroids which are not of the form $M(G, \Psi)$. For example, in the following graph $Q$ we can define an equivalence relation on the (edge sets of) double rays by saying that two double rays are equivalent if they have finite symmetric difference. Then the set of finite circuits together with any collection of double rays closed under this equivalence relation gives the set of circuits of an infinite matroid. The matroids I just described are a bit pathological, and they hover on the boundary between being binary and non-binary. None of them has a $U_{2,4}$-minor. They also still have the familiar property that any symmetric difference of two circuits is a disjoint union of circuits. But symmetric differences of three circuits might not be disjoint unions of circuits!

For example, there is such a matroid in which the first three sets depicted below are circuits, but the fourth, their symmetric difference, is not. The problem is that these matroids are wild. This means that there are circuits and cocircuits which intersect in infinitely many elements. We only have a good theory of representability for tame matroids, those in which every intersection of a circuit with a cocircuit is finite. I hope to discuss this in more detail in a future post.

If we only consider tame $G$-matroids, then this proliferation of pathological matroids disappears. For the graph $Q$, for example, there are only 2 tame $Q$-matroids, namely the finite cycle matroid and the topological cycle matroid. Remarkably, for any locally finite graph $G$ it turns out that the tame $G$-matroids are precisely the matroids of the form $M(G, \Psi)$. So our top-down and bottom up characterisations meet, and any matroid associated to a graph can be built up from finite parts in a way which mirrors the structure of that graph. The reasons for this correspondence go far beyond the scope of this post, but they can for example be found in [BC16].

Now that we’ve seen the standard ways to build infinite matroids and their relationship to infinite graphs, in the next post we’ll examine the most important open problem about them: the Infinite Matroid Intersection Conjecture.

[BC15] N. Bowler and J. Carmesin, Infinite Matroids and Determinacy of Games, preprint here.
[BC16] N. Bowler and J. Carmesin, The ubiquity of Psi-matroids, preprint here.
[D10] R. Diestel, Locally finite graphs with ends: a topological approach I–III, Discrete Math 311–312 (2010–11).
[G07] A. Georgakopoulos. Connected but not path-connected subspaces of infinite graphs, Combinatorica, 27(6) 683–698 (2007).