# Finding Shortest Circuits in Binary Matroids

Two popular invariants in graph theory are the girth of graph (the size of its smallest cycle) and its edge-connectivity. Given a graph, both can be easily computed. In fact, if $e$ is an edge and $s$ and $t$ are its ends, then finding the shortest cycle or smallest edge-cut containing $e$ can be done by finding a shortest $s,t$-path or a minimum $s,t$-cut, both easy problems.

Can we generalize the problem of computing these invariants to matroids? Cycles and minimal edge-cuts in graphs correspond to circuits and cocircuits in their cycle matroids (as long as we count loops and pairs of parallel edges as cycles). So we know how to find a shortest circuit or cocircuit in a graphic matroid. Note that in the matroid world, these become just one problem, since computing the smallest size of a cocircuit is the same under duality as computing the girth of a matroid, the smallest size of a circuit.

In this post I will discuss computing the girth of a matroid. We will only consider binary matroids, since the problem is already intractable just for this class: Alexander Vardy  proved that it is NP-hard to compute the girth of a given binary matroid (he actually worked on an equivalent problem from coding theory, that of finding the minimum distance of a binary linear code).

But, we know that it is easy to compute the girth of two types of binary matroids, the graphic and cographic ones. Are there other subsets of the binary matroids where the problem is still tractable? The class of regular matroids is an obvious candidate to consider after the graphic and cographic. In fact, the regular matroid decomposition theorem of Paul Seymour says that any regular matroid can be built by repeated $1$-, $2$-, and $3$-sums of graphic matroids, cographic matroids, and one particular matroid called $R_{10}$. This means that for any regular matroid $M$, there is a tree $T$, each of whose vertices is either a graphic matroid, a cographic matroid, or a copy of $R_{10}$, and such that $M$ is built by $(\leq 3)$-sums from these smaller matroids, each sum corresponding to an edge of the tree. Computing the girth of regular matroids is easy if we have an algorithm to find such a tree. How to find it is beyond the scope of this post, but more information can be found in .

Now let’s see how to compute the girth of a regular matroid. It’s actually easier if we allow matroid elements to be weighted. If each element $e$ of a matroid $M$ has a non-negative weight $w(e)$, then we’ll say that the girth of $M$ is $\min\left\{\sum_{e \in C} w(e) : C \text{ is a circuit of } M\right\} .$ For graphic and cographic matroids, this introduces no difficulties since we can find minimum cuts and shortest paths even in weighted graphs.

Theorem 1. There is a polynomial-time algorithm to compute the girth of a given regular matroid $M$ with non-negative weights on its elements.

Proof. First, we need to get the tree $T$ mentioned above. If $T$ has one vertex, then $M$ is just graphic, cographic, or a copy of $R_{10}$, and the problem is easy. Otherwise, we pick any leaf of the tree, $v$. So $v$ corresponds to a matroid $M_1$ that is either graphic, cographic, or a copy of $R_{10}$. The tree $T – v$ corresponds to a regular matroid $M_2$, such that $M$ is a $(\leq 3)$-sum of $M_1$ and $M_2$. If $M$ is a $1$-sum of $M_1$ and $M_2$, then the girth of $M$ is the minimum of the girths of $M_1$ and $M_2$, so we can find it by induction.

Otherwise, suppose that $M$ is a $2$-sum of $M_1$ and $M_2$. Denote by $e$ the common element in $E(M_1)$ and $E(M_2)$. We give each element of $M_1$ and of $M_2$ the same weight as it had in $M$, and we give $e$ a weight of zero in $M_1$. We will assign the weight of $e$ in $M_2$ later. Since $M_1$ is either graphic, cographic, or a copy of $R_{10}$, we can find the following two things in polynomial time: the first is a minimum-weight circuit $C$ of $M_1 \backslash e$ and the second is a minimum-weight circuit $C’$ of $M_1$ containing $e$.

We now assign the value $\sum_{f \in C’} w(f)$ as the new weight of $e$ in $M_2$. It is not too hard to see that the girth of $M$ is the minimum of the weight of $C$ and the girth of $M_2$. We can therefore find it by induction.

The last case is when $M$ is a $3$-sum of $M_1$ and $M_2$. This is just a variant of the $2$-sum case but we need to find four circuits in $M_1$ instead of two; the details are omitted. $\square$

We have seen that computing girth is easy even in regular matroids, and the reason was that we can reduce them to graphic and cographic matroids. What about other classes of binary matroids? In an earlier entry on this blog, Peter Nelson talked about a theorem announced by Jim Geelen, Bert Gerards and Geoff Whittle  which gives a decomposition of binary matroids in any proper minor-closed class similar to that given by the decomposition theorem for regular matroids. The pieces in this decomposition are actually not that different from graphic and cographic matroids.

If $N$ is a binary matroid and $A$ is a matrix such that $N = M(A)$, then a rank-($\leq t$) perturbation of $N$ is a matroid of the form $M(A + T)$, where $T$ is a binary matrix with rank at most $t$.

Here is the statement of their theorem (Theorem 3.1 of ) from that earlier post:

Theorem 2. Let $\mathcal{M}$ be a proper minor-closed class of binary matroids. There exist integers $k$ and $t$ so that every vertically $k$-connected matroid in $\mathcal{M}$ is a rank-($\leq t$) perturbation of a graphic or cographic matroid.

This means that, for any proper minor-closed class $\mathcal{M}$ of binary matroids, we can make an efficient algorithm to compute the girth of members of $\mathcal{M}$ if we can do two things efficiently:

1. Compute the girth of a given rank-($\leq t$) perturbation of a graphic or a cographic matroid, and
2. Reduce the problem of computing the girth of members of $\mathcal{M}$ to that of computing the girth of vertically $k$-connected members of $\mathcal{M}$.

We don’t yet know how to solve either of these two subproblems in polynomial time, though in joint work Jim Geelen and I have mostly solved the first one by finding randomized algorithms to do it, which run in polynomial time and work correctly with probability nearly 1 (I won’t say more about those algorithms in this post though).

When $\mathcal{M}$ is the class of regular matroids, Theorem 2 holds with $k = 4$ and $t = 0$ (ignoring the slight exception of $R_{10}$). We can thus compute girth in regular matroids by reducing to the easy case of graphic and cographic matroids.

Unfortunately, the trick we used to do this reduction for regular matroids does not work so easily for other classes. For regular matroids, we needed to be able to find a minimum-weight circuit in a graphic or cographic matroid that used a specified element. While this was easy, the requirement of using a specified element makes things trickier for perturbations.

In the remainder of this post, I will outline a way to compute girth only for one special type of perturbation of graphic matroids, known as the even-cut matroids. But since it’s the same problem under matroid duality, we’ll actually look at cogirth (the smallest size of a cocircuit) in the duals of even-cut matroids, which are projections of graphic matroids. A projection of a graphic matroid $M(G)$ means a matroid of the form $M(A) / e$, where $A$ is a binary matrix obtained by adding one new column, indexed by an element $e$, to the vertex-edge incidence matrix of a graph $G$. The cocircuits of this matroid are precisely the cocircuits of $M(A)$ disjoint from $e$. But we need a way to describe these cocircuits in terms of the graph $G$.

We can view the new column of $e$ that we added to the vertex-edge incidence matrix of $G$ as the characteristic vector of a set of vertices of $G$. So we can fully describe this matroid by the pair $(G, T)$ where $T$ is a subset of $V(G)$. We can assume that $|T|$ is even; if not, then $e$ is a coloop of $M(A)$ and so $M(A) / e$ is graphic and the problem becomes easy.

Now we can describe the cocircuits in terms of the graph: they are precisely the minimal edge-cutsets of the form $\delta(X)$ where $|X \cap T|$ is even. Let’s call such edge-cutsets $T$-even cuts. How can we find a minimum-size $T$-even cut and thereby compute the cogirth of this projection of a graphic matroid? Here is a solution from Barahona and Conforti .

Let’s say that $\delta(W)$ is a minimum $T$-even cut in the graph $G$, so we want to either find it, or find another one of equal size. We already know how to find a minimum cut in $G$ in poylnomial time, so we’ll find one of those and call it $\delta(U)$. If we are lucky and $|U \cap T|$ is even, then we’re done. Otherwise, $|U \cap T|$ is odd.

Now let’s look at the possible interactions between $W$ and $U$. Exactly one of the sets $|W \cap U \cap T|$ and $|(V(G) – W) \cap U \cap T|$ is odd and the other is even; by the symmetry between $W$ and $V(G) – W$, let’s say that $|W \cap U \cap T|$ is even. And since $|W \cap T|$ is even, the set $|W \cap (V(G) – U) \cap T|$ is also even.

That means that $\delta(W \cap U)$ and $\delta(W \cap (V(G) – U))$ are also $T$-even cuts! Unless they are empty. But they can’t both be empty, or $W$ would be empty. So suppose, by symmetry, that $\delta(W \cap U)$ is a non-empty $T$-even cut. Then we apply the submodularity inequality: $|\delta(W \cap U)| + |\delta(W \cup U)| \leq |\delta(U)| + |\delta(W)|.$ Since $\delta(U)$ is a minimum cut, we have $|\delta(U)| \leq |\delta(W \cup U)|$, which means that $|\delta(W \cap U)| \leq |\delta(W)|$ so $\delta(W \cap U)$ is actually a minimum $T$-even cut!

So we’ve shown that there is a minimum $T$-even cut of the form $\delta(W’)$, where $W’$ is either contained in $U$ or contained in $V(G) – U$. We construct two graphs: $G_1$ is obtained from $G$ by identifying all vertices in $U$ to a single vertex, and $G_1$ is obtained by identifying all vertices in $V(G) – U$ to a single vertex. In both cases, we add the new vertex to the set $T$. We can check that every $T$-even cut of $G_1$ or of $G_2$ is also a $T$-even cut of $G$. But in addition, our desired cut $\delta(W’)$ is definitely a $T$-even cut in either $G_1$ or $G_2$. So we finish by recursively solving the problem for these two smaller graphs.

# References

 Francisco Barahona and Michele Conforti, A construction for binary matroids, Discrete Math. 66 (1987), 213-218.

 Jim Geelen, Bert Gerards and Geoff Whittle, The highly connected matroids in minor-closed classes, arXiv:1312.5012 [math.CO].

 K. Truemper, Matroid Decomposition, Leibniz, Plano, Texas, 1998.

 Alexander Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory 43 (1997), 1757-1766.

# Matroids over partial fields from graphs embedded in surfaces

Guest post by Daniel Slilaty.

In a series of three posts by Irene Pivotto (on 08-26-13, 10-07-13, and 10-28-13) subscribers to The Matroid Union blog were introduced to biased graphs and their matroids. Again, a biased graph is a pair $(G,B)$ in which $G$ is a graph and $B$ is a collection of cycles in $G$ (called balanced) for which any theta subgraph of $G$ contains 0, 1, or 3 balanced cycles, i.e. not exactly two balanced cycles. (A theta subgraph is the union of a pair of cycles that intersect along a path.)

Biased graphs were first defined by Zaslavsky [Za91] and they are primarily used to characterize two types of matroids: single-element coextensions of graphic matroids and frame matroids [Za94]. Frame matroids contain the very important class of Dowling Geometries whose centrality within matroid theory was first displayed by Kahn and Kung [KK82] and more recently by Geelen, Gerards, and Whittle [GGW14]. Given a biased graph $(G,B)$, denote the frame matroid by $F(G,B)$ and the complete lift matroid by $L_0(G,B)$ (i.e., the single-element coextension of $M(G)$ defined by $B$).

The canonical examples of biased graphs are given by gains over a group. Formally, this is a function $\varphi\colon\vec E(G)\to\Gamma$ where $\vec E(G)$ is the collection of oriented edges of a graph (if $e$ is an oriented edge, then $-e$ is the same underlying edge oriented in the opposite direction) and $\Gamma$ is a group such that $-\varphi(e)=\varphi(-e)$ for additive groups and $\varphi(e)^{-1}=\varphi(-e)$ for multiplicative groups. In this post we will only consider abelian groups. One advantage of abelian groups over nonabelian ones is that gain functions, up to a certain type of equivalence, gives rise to and can be specified by homomorphisms $\hat\varphi\colon Z_1(G)\to\Gamma$ in which $Z_1(G)$ is the integer cycle space of the graph. Given $\varphi$, let $B_\varphi$ be the collection of cycles in $G$ for which $\hat\varphi(\vec C)$ is the identity element of $\Gamma$ (0 when $\Gamma$ is additive, 1 when $\Gamma$ is multiplicative). Now $(G,B_\varphi)$ is a biased graph.

Given a biased graph $(G,B)$, a homomorphism $\hat\varphi\colon Z_1(G)\to\Gamma$ such that $B_\varphi=B$ is called a $\Gamma$-realization of $(G,B)$. The following proposition by Zaslavsky can be stated for skew fields as well as ordinary fields.

Proposition 1 (Zaslavsky [Za03]) Let $(G,B)$ be a biased graph and $\mathbb F$ a field.

1. If $(G,B)$ is realizable over the multiplicative subgroup of $\mathbb F$, then $F(G,B)$ is $\mathbb F$-representable.
2. If $(G,B)$ is realizable over the additive subgroup of $\mathbb F$, then $L_0(G,B)$ is $\mathbb F$-representable.

Partial fields have become a central and popular topic in modern matroid theory. (See Stefan van Zwam’s blog post from 09-16-13 for an introduction to partial fields.) Briefly a partial field is a pair $\mathbb P=(R,U)$ in which $R$ is a commutative ring and $U$ is a subgroup of the multiplicative group of units of $R$ such that $-1\in U$.

Proposition 2 (van Zwam [vZ09]) Let $\mathbb P=(R,U)$ be a partial field and $(G,B)$ a biased graph.

1. Let $\varphi\colon\vec E(G)\to U$ be a $U$-realization of $(G,B)$. If $\varphi(\vec C)$ is a fundamental element of $\mathbb P$ for each cycle $C$ of $G$, then the frame matroid $F(G,B)$ is $\mathbb P$-representable.
2. Let $\varphi\colon\vec E(G)\to R_+$ be an $R_+$-realization of $(G,B)$. If $\varphi(\vec C)\in U$ for every unbalanced cycle $C$ of $(G,B)$, then $L_0(G,B)$ is $\mathbb P$-representable.

“Well-connected” examples of matroids representable over various partial fields can sometimes be hard to come by. The largest possible simple $\mathbb P$-matroids of a given rank are known for various partial fields, but not every $\mathbb P$-matroid of a given rank must be contained within a $\mathbb P$-matroid of maximum size. Sometimes biased graphs defined by embeddings in surfaces can provide interesting examples of $\mathbb P$-matroids as well. Given a graph $G$ embedded in a closed surface $S$, invariance of homology gives us a natural homomorphism $\natural\colon Z_1(G)\to H_1(S)$ where $H_1(S)$ is the first homology group of the surface calculated with integer coefficients. Now given a partial field $\mathbb P=(R,U)$ there may be some choice of $\sigma\colon H_1(S)\to\mathbb P$ such that the biased graph $(G,B_{\sigma\natural})$ satisfies the conditions of Proposition 2.

Two of my favorite examples come from graphs embedded in the Klein Bottle. The Klein bottle is the surface obtained by identifying two Möbius bands along their circular boundary. In the figure below we have the lighter and darker Möbius bands. The first homology group $H_1(K)$ of the Klein bottle is $\mathbb Z\times\mathbb Z_2$. Of particular usefulness for us is the fact that any cycle $C$ embedded in the Klein bottle now has $\pm\natural(\vec C)\in\{(0,0),(1,0),(0,1),(1,1),(2,0)\}$. Now let $G$ be a nice large graph embedded in the Klein bottle with high face width e.g., a square grid.

• The dyadic partial field $\mathbb D=(\mathbb Q,U)$ has $U=\{\pm 2^i:i\in\mathbb Z\}$. Define $\sigma\colon\mathbb Z\times\mathbb Z_2\to\mathbb Q_+$ by $(1,0)\mapsto 1$ and $(0,1)\mapsto 0$. This yields $\sigma(a,b)\in\{0,\pm 1,\pm 2\}\subset U$ and so $L_0(G,B_{\sigma\natural})$ is $\mathbb D$-representable by
Proposition 2 (2).
• The 2-cyclotomic partial field $\mathbb K_2=(\mathbb Q(x),U)$ has $U=\{\pm x^i(1-x)^j(1+x)^k:i,j,k\in\mathbb Z\}$. Define $\sigma\colon\mathbb Z\times\mathbb Z_2\to U$ by $(1,0)\mapsto x$ and $(0,1)\mapsto \pm1$. Either choice for $\sigma(0,1)$ yields $\sigma(a,b)\in\{1,x,-x,x^{-1},-x^{-1},x^2,x^{-2}\}$ and each element of this set is a fundamental element of $\mathbb K_2$ [vZ09]. Thus $F(G,B_{\sigma\natural})$ is $\mathbb K_2$-representable by Proposition 2(1). Representations of matroids over $\mathbb K_2$ have homomorphic images over other interesting partial fields as well, e.g., the golden-mean and Gersonides partial fields.

These are two very nice examples, but are there more? Do other surfaces yield interesting partial fields to work with? Other surfaces have restrictions on the homology classes of cycles embedded on their surface although these restrictions are not as tight as those for the Klein bottle. The first homology group for the torus is $\mathbb Z\times\mathbb Z$ and cycles must be in homology class $(a,b)$ where the greatest common divisor of $a$ and $b$ is 1. The first homology group of Dyck’s surface (i.e., the connected sum of the torus and the projective plane) is $\mathbb Z\times\mathbb Z\times \mathbb Z_2$ and cycles must be in homology classes $(a,b,0)$, $(a,b,1)$, or $(2a,2b,0)$ where the greatest common divisor of $a$ and $b$ is 1. Then again, is it necessary to restrict ones attention to graphs embedded in surfaces? There are other interesting topological spaces that could be explored as well: dunce hats and connected sums of surfaces with dunce hats, for example.

References

[KK82] J. Kahn, J. Kung, Varieties of combinatorial geometries, Trans. Amer. Math. Soc. 271 (1982) 485–499.

[GGW14] J. Geelen, B. Gerards, G. Whittle, Solving Rota’s conjecture, Notices Amer. Math. Soc. 61 (2014) 736–743.

[vZ09] S. van Zwam, Partial Fields in Matroid Theory, Doctoral dissertation, Technische Universiteit Eindhoven, Netherlands, 2009.

[Za89] T. Zaslavsky, Biased graphs. I. Bias, balance, and gains, J. Combin. Theory Ser. B 47 (1989) 32–52.

[Za91] T. Zaslavsky, Biased graphs. II. The three matroids, J. Combin. Theory Ser. B 51 (1991) 46–72.

[Za94] T. Zaslavsky, Frame matroids and biased graphs, European J. Combin. 15 (1994) 303–307.

[Za03] T. Zaslavsky, Biasedgraphs IV: Geometrical realizations, J. Combin. Theory Ser. B 89 (2003) 231–297.

# Infinite matroids

Guest post by Reinhard Diestel.

This is the first of what might become an irregular series of posts on infinite matroids, written on the invitation of Irene Pivotto for the Union. This first piece tells the story of how finite matroids and infinite graphs conspired to get us to think about how to axiomatize infinite matroids. How we ended up solving an old problem we had not even known about – and how it came to be that infinite matroid theory is suddenly exploding. It is meant to be light reading, with some deeper hints to ponder. Future posts, to be written by Nathan Bowler, are likely to be more mathematical and less anecdotal, as they follow up the various open ends of the emerging theory.

Back in 1966, Rado  asked whether there was a way to axiomatize matroids in such a way that infinite matroids, too, would have duals. A big challenge, given what was known. But we knew none of this.

What we did know was graphs. In particular, infinite graphs . And these were luring us – towards matroids.

It had all begun with a dream, many years earlier. As an undergraduate studying with Halin, I had heard of the ends of an infinite graph: limit points at infinity such as the three red dots in this graph: There is a combinatorial definition of ends that divorces the notion of convergence appealed to in this example from the topology of the plane: an end is taken to be an equivalence class of rays, that is, 1-way infinite paths, where two rays are considered as equivalent if no finite set of vertices separates them. As is easy to check, the graph above has three such classes each of which corresponds to one of the red dots, in that its rays converge to that dot in the plane.

Now here was my dream, inspired by this picture. There are a number of theorems about cycles in finite graphs that do not readily generalize to infinite graphs; Hamilton cycles are an obvious example. Might infinite cycles’ such as the perimeter circle of the above graph come to the rescue and give us infinite analogues of finite cycle theorems? In our example, the perimeter circle would be an infinite Hamilton cycle. And the formal definition of an infinite cycle would be: a cyclic sequence of alternately double rays and ends, with is an element of’ as incidence. What fun!

But when I tried this in earnest several years later, with my student Daniela Kühn, we soon got bogged down. It turned out that infinite cycles such as the above did not always
work: we had to iterate to allow cycles like Then for the same reasons we had to iterate again, iterate transfinitely – and ultimately failed. The end of the story  was that topology, of all things, came to the rescue.
Even for graphs with abstract ends, not necessarily planar, there is a natural topology in which rays converge to their’ ends. And it was the edge sets of circles in this topology that turned out to be the right notion of infinite cycle in locally finite infinite graphs, in the sense of allowing us to generalize those finite graph theorems. But these circles could be wild: containing infinitely many double rays arranged like the rationals, and continuum many ends! The perimiter circle of the planar graph below is an example: And now matroids began to call. One of the finite graph theorems that wouldn’t generalize to infinite graphs naively was the tree packing theorem of Nash-Williams and Tutte. But we were able to prove an infinite tree packing theorem for the naturally adapted topological notion of tree: a set of edges that connects all the vertices but whose topological closure does not contain a circle. Shouldn’t there be a notion of infinite matroid hiding behind this, one in which these topological trees would be the bases, and edge sets of topological circles the circuits?

The usual matroid axioms, together with

(I4) An infinite set is independent as soon as all its finite subsets are independent

as known from vector spaces, do not cover such matroids: a circuit, with all its proper subsets independent, can obviously not be infinite. So we looked for new axioms: axioms for infinite matroids that would default to the known finite ones for finite structures, but which would also allow for infinite circuits such as those topological ones.

It was fun trying – slowly, experimenting as much as thinking, and unaware that others had tried before us. Eventually, we came up with five sets of infinite matroid axioms  – in terms of independent sets, bases, circuits, closure and rank – that worked in the way I had hoped: they made the edge sets of topological circles of a locally finite infinite graph into the circuits of a matroid, and our topological spanning trees into the bases of that matroid. (In due course, even a base packing theorem emerged that implied our topological tree packing theorem, as hoped for . But that was much later.) Thus, for an infinite graph there were now two different cycle matroids: one whose circuits were the finite cycles, and another whose circuits were the edge sets of topological circles, finite or infinite. And each of these had a dual: the cocircuits of the first were all the bonds (finite or infinite), those of the latter precisely the finite bonds .

In particular: it turned out, with hindsight, that those infinite cycles’ which had given us such pain to find, and which we were able to nail down only with the help of topology, had a combinatorial description after all: as the minimal non-empty sets of edges not meeting any finite bond in just a single edge. It was thus the matroids lurking in the background which – once found – provided that much-sought combinatorial description of infinite cycles that would make theorems about cycles in finite graphs generalize to infinite graphs!

To top it all, it turned out that we had solved that ancient problem of Rado’s. Or more precisely: that we discovered that Higgs and Oxley had solved it long before us. But this is a story for a later blog, in which we shall look at the infinite axioms more explicitly and give more concrete examples of infinite matroids.

To wind up this one, let me say a little about that traditional axiom (I4): why it was a natural one to suggest at the time matroids were invented, but also why it is – from a modern perspective – a somewhat simple-minded one.

Axiom (I4) makes independence into a property of an infinite combinatorial structure that depends only on its finite substructures. Such properties can typically be verified by an application of Zorn’s Lemma, or by a so-called compactness argument: one that uses the compactness of an infinite product of compact spaces, typically finite, as proved by Tychonoff in the 1930s. Compactness was en vogue at the time and emerged in various guises, and so it must have been natural then to specify the infinite independent sets of a matroid as determined by the finite ones as in (I4).

But let’s look at this problem again now, even put in this way: given the collection of finite subsets of some given set that we wish to call independent, which infinite subsets should we grant the same status?

Axiom (I4) gives a straightforward answer: take them all. All, that is, that have a chance of complying with that other intended axiom, that subsets of independent sets shall be independent. And Rado, one of the champions of compactness and other equivalents of the axiom of choice, saw that this was too crude: in order for infinite matroids to allow for duality as we know it from finite ones, we have to choose as the infinite independent sets a carefully balanced subclass of all those allowed by (I4). To find this subclass, and to describe it by axioms both subtle and simple enough to yield an interesting theory of infinite matroids with duality, was his 1966 challenge.

It now seems that we are finally there. With those newly found axioms, infinite matroid theory can at last be developed in line with its finite counterpart that has raced so successfully ahead. The train is already gaining momentum, inviting you to jump on – but this, too, will be a story for later posts.

 E. Aigner-Horev, J.-O. Fröhlich and J. Carmesin, Infinite matroid union, arXiv:1111.0602.

 H. Bruhn, R. Diestel, M. Kriesell, R. Pendavingh and P. Wollan, Axioms for infinite matroids, Adv. Math 239 (2013), 18–46.

 H. Bruhn and R. Diestel, Infinite matroids in graphs, in the Infinite Graph Theory special volume of Discrete Math. 311 (2011), 1461–1471.

 R. Diestel, The cycle space of an infinite graph, Comb. Probab. Computing 14 (2005), 59–79.

 R. Diestel, Locally finite graphs with ends: a topological approach I–III, Discrete Math 311–312 (2010–11).

 R. Rado, Abstract linear dependence, Colloq. Math. 14 (1966), 257–64.