This post is about some conjectures on matroid minor structure theory in the most general setting possible: excluding a uniform matroid. Jim Geelen previously discussed questions in this area in [1], and most of what I write here comes from discussions with him.

Whether they be for graphs or matroids, qualitative structure theorems for minor-closed classes usually fit a particular template. They make a statement that the members of a minor-closed class are ‘close’ to having a particular basic structure, where the structure should be something that differs the graphs/matroids in the class from arbitrary/random ones. In the case of proper minor-closed classes of graphs, this structure takes the form of embeddability on a fixed topological surface, as was famously shown by Robertson and Seymour. For matroids representable over a fixed finite field, the structure arises from classes of group-labelled graphs that embed on a fixed topological surface, classes of group-labelled graphs in general, and (when the order of the field is non-prime) classes of matroids representable over a fixed subfield. These results for finite fields have been shown in recent years by Geelen, Gerards and Whittle, and have already given us powerful tools for understanding the matroids in minor-closed classes, as demonstrated most notably in their proof of Rota’s conjecture.

Extending graph minors structure theory to minor-closed classes of matroids over finite fields was no mean feat, but there is real reason to believe that we can generalise it further. It seems unlikely that one could obtain very strong structural results for *all* minor-closed classes; for example, the class of *sparse paving matroids* (that is, the matroids whose girth is at least the rank and whose cogirth at least the corank) is minor-closed, and is conjectured to contain almost all matroids – intuitively, a structure theorem that allows us to ‘understand’ the class of sparse paving matroids seems unlikely. The sticking point here seems to be that this class contains all uniform matroids. At the moment, we think that one should be able to obtain sensible qualititative structure theorems precisely for the minor-closed classes that don’t contain all the uniform matroids. In this post, I’ll state a conjectures that makes this kind of statement. For this, we need two ingredients.

**Basic Structures**

We first need an idea which basic structures we expect to see as outcomes. These turn out to be pretty close to what goes on in the finite field representable case. Let $\mathcal{M}$ be a minor-closed class not containing all uniform matroids. Let $U$ be a uniform matroid not contained in $\mathcal{M}$. Without loss of generality, we may assume that $U = U_{s,2s}$ where $s \ge 4$, as every uniform matroid is a minor of a matroid of this form. Here are some examples of interesting classes that $\mathcal{M}$ could be:

- The class of $\mathbb{F}$-representable matroids, where $\mathbb{F}$ is some finite field over which $U$ is not representable.
- The class of graphic matroids.
- The class of bicircular matroids. (These are the ‘graph-like’ matroids in which each edge element is placed freely on the line between two vertex elements).
- The class of frame matroids (i.e. the matroids of the form $M \backslash V$, where $V$ is a basis of a matroid $M$ for which every fundamental circuit has size at most two). This class contains both the previous ones, as well as various other graph-like matroids such as those arising from group-labelled graphs. $U_{4,8}$ is not a frame matroid and therefore neither is $U$.
- The class of duals of frame matroids. (Or graphic/bicircular matroids)
- A class of matroids arising (as graphic, cographic, bicircular, group-labelled-graphic, etc…) from the class of graphs that embed on some surface.

Although they are not at all trivial from a graph-theoretic perspective, classes of the last type are less rich as they all have small vertical separations. Classes of the the other types, however, contain matroids of arbitrarily high vertical connectivity. Structural statements simplify a lot when they are made about highly connected matroids, and the main conjecture I’ll be stating will apply only to very highly vertically connected matroids in a given minor-closed class; the last type of class will therefore not show up.

The divide between the ‘highly connected’ minor-closed classes and other classes is made concrete by the following conjecture.

**Conjecture 1. **Let $\mathcal{M}$ be a minor-closed class of matroids not containing all uniform matroids. Either

- $\mathcal{M}$ contains the graphic matroids or their duals,
- $\mathcal{M}$ contains the bicircular matroids or their duals, or
- there is an integer $t$ so that $\mathcal{M}$ does not contain any vertically $t$-connected matroid on at least $2t$ elements.

**Closeness **

We now need to say what, according to our structural conjecture, exactly is meant by two matroids being ‘close’. For minor-closed classes of graphs, ‘closeness’ is measured in terms of vortices, apex vertices, and clique-sums. For matroids over finite fields, the structure theory considers two matroids on the same ground set to be ‘close’ if they have representations that differ by a bounded-rank matrix. None of these ideas quite work for matroids in general, as we don’t have representations, let alone vertices. However, there is something that will.

A *single-element **projection *of a matroid $M$ is a matroid $N$ for which there exists a matroid $L$ such that $L \backslash e = M$ and $L / e = N$. If $N$ is a single-element projection of $M$ then $N$ is a *single-element lift *of $M$. Single-element lifts and projections are dual notions that ‘perturb’ a matroid by a small amount to another matroid on the same ground set; for example, they change the rank and corank of any given set by at most $1$. We write $\mathrm{dist}(M,N)$ to denote the minimum number of single-element lifts and/or projections to transform $M$ into $N$. If this ‘distance’ is small, then $M$ and $N$ are ‘close’. This will serve as the notion of closeness in our structure conjecture. (In a more general structure theory, vortices will also arise). This distance only makes sense if $M$ and $N$ have the same ground set, if this is not the case, then we say $\mathrm{dist}(M,N) = \infty$.

Incidentally (and I’m looking at you, grad students), I would like an answer to the following question, which would simplify the notion of perturbation distance – I don’t have a good intuition for whether it should be true or false, and either way it could be easy.

**Problem. **Let $M$ and $N$ be matroids for which there exists a matroid $L$ that is a single-element lift of both $M$ and $N$. Does there exist a matroid $P$ that is a single-element projection of both $M$ and $N$?

**The Conjecture**

Here goes! If $\mathcal{M}$ is a minor-closed class not containing all uniform matroids, then the highly connected members of $\mathcal{M}$ should be close to being frame, co-frame, or representable.

**Conjecture 2: **Let $\mathcal{M}$ be a minor-closed class of matroids not containing a uniform matroid $U$. There is an integer $t \ge 0$ so that, if $M$ is a vertically $t$-connected matroid in $\mathcal{M}$, then there is a matroid $N$ such that $\mathrm{dist}(M,N) \le t$ and either

- $N$ is a frame matroid,
- $N^*$ is a frame matroid, or
- $N$ is representable over a field $\mathbb{F}$ for which $U$ is not $\mathbb{F}$-representable.

If true, this conjecture would tell us that minor-closed classes of matroids inhabit a very beautiful universe. The hypotheses are very minimal, applying to a massive variety of different $\mathcal{M}$. The conclusions, on the other hand, imply that all ‘nondegenerate’ members of $\mathcal{M}$ are at a small distance from a matroid $N$ that is *very *far from being generic. Somehow, as happens again and again in matroid theory, the ‘special’ classes of representable and graph-like matroids are in fact fundamental to understanding the minor order.

Conjecture 2 is one of the many goals of a more general matroid structure theory; in my next post I will discuss some recent progress we have made towards it.

**Reference**

[1] *Some open problems on excluding a uniform matroid*, J. Geelen, Adv. Appl. Math. 41 (2008), 628-637.

About the problem, or I missed something very basic or the answer is “iff M=N”:

If N is a single element projection of M, then either M=N (e is a loop or coloop) or r(N)=r(M)-1.

For distinct M and N in the problem we would have: r(M)-1<=r(L)<=r(N)+1 with one of the inequalities being strict because we don't have M=L=N. So, r(M)<r(N). But, if such L' exists, we have r(N)<=r(L')<=r(M).

Oh, I’m sorry, I stated it wrong. It will be fixed in a second.

I think I have a negative answer to the problem. Let L be the cube, represented by the vectors {(v,1): v in {0,1}^3} over the reals. Let N’ be the extension of L by a free element e and M’ an extension of L by the vector f=(0,0,1,0). Note that N:=N’/e is isomorphic to U_{3,8} and M:=M’/f is obtained by adding a parallel element to each element of U_{3,4}. Each single-element projection of M has all its elements in circuits with at most two elements. But this never happens to a single-element projection of N.

Nevermind, that is wrong.

Think I got it right this time:

Let L have ground set {0,1,2}x{0,1}x{0,1}x{1} with the notion of linear dependence over the reals. Define:

M:=(L+(1,0,0,0))/(1,0,0,0) and

N:=(L+(0,0,1,0))/(0,0,1,0).

Note that M (resp. N) is obtained from U_{3,4} (U_{3,6}) by replacing each element by a parallel class with three (resp. two) elements.

Each single-element projection M’/e of M (resp. N’/e of N) either:

-is M (resp. N), when e is a loop or coloop;

-is obtained from U_{2,4} (resp. U_{2,6}) by replacing each element by a parallel class with three (resp. two) elements, when e is free;

-has at least one parallel class with six (resp. four) elements, no loops, and other parallel classes having three or six elements (resp. two or four elements), when e is in some triangle of M’ (resp. N’); or

-has exactly three (resp. two) loops, when e is in a parallel pair of M’.

This implies that M and N has no common single-element projection.

sorry, got it wrong again. N is not obtained this way. Think I need some sleep.

Yeah, it can be tricky to think of examples for this one. My intuition is that one shouldn’t be able to construct counterexamples in this way because of the representability – in particular, if M = (L + e)/e and N = (L + f)/f, and the matroid L + e + f is well-defined, then (L + e + f)/{e,f} will be a common projection. This will happen in the representable case if L has a representation that can be extended by both e and f. I would predict that a counterexample would be ‘essentially non-representable’ in some way, or at least there would have to be issues with uniqueness of representability.

Noticed the thing with the representability. Now I`m trying a positive answer. I think the following outline might work:

We have modular cuts C_1 and C_2 in L that induces single-element projections M_1 and M_2. For {j,k}={1,2} we define a modular cut B_k of M_k as the set of flats of M_k in C_j (we have that each flat of M_i is a flat of L). Let P_k be the single element projection of M_k induced by B_k. I conjecture that P_1=P_2. I’ll try to make the details later.

Well, I made the details. It works provided such B_k is indeed a modular cut. But I failed to prove this part. Seems like a clue about how a counter-example would look like.