Related papers: The smallest matroids with no large independent fl…

The configuration of a matroid $M$ is the abstract lattice of cyclic flats (flats that are unions of circuits) where we record the size and rank of each cyclic flat, but not the set. One can compute the Tutte polynomial of $M$, and stronger…

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for…

We show that if the ground set of a matroid can be partitioned into $k\ge 2$ bases, then for any given subset $S$ of the ground set, there is a partition into $k$ bases such that the sizes of the intersections of the bases with $S$ may…

A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such…

Let $s,n \ge 2$ be integers. We give a qualitative structural description of every matroid $M$ that is spanned by a frame matroid of a complete graph and has no $U_{s,2s}$-minor and no rank-$n$ projective geometry minor, showing that every…

We construct a family of independent sets for finite, atomic, and graded lattices, extending the well-known cryptomorphism between geometric lattices and matroids. This construction leads to an embedding theorem into geometric lattices that…

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank…

The critical threshold of a (simple binary) matroid $N$ is the infimum over all $\rho$ such that any $N$-free matroid $M$ with $|M|>\rho2^{r(M)}$ has bounded critical number. In this paper, we resolve two conjectures of Geelen and Nelson,…

A frame template over a field $\mathbb F$ describes the precise way in which a given $\mathbb F$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is…

In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has $K_5$ or $K_{2,2,2}$ as a minor. Mills and Turner proved an analog of this theorem by showing that every $3$-connected binary matroid in which…

Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb{P}$, let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary, and suppose $M$ has a pair of elements…

For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected…

Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle,…

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…

We give an excluded-minor characterization for the class of matroids M in which M\e or M/e is binary for all e in E(M). This class is closely related to the class of matroids in which every member is binary or can be obtained from a binary…

We show that a determinantal ideal generated by $t$-minors does not contain any nonzero polynomials with $t!/2$ or fewer terms. Geometrically this means that any nonzero polynomial vanishing on all matrices of rank at most $t-1$ has more…

In this paper, we give a complete characterization of binary matroids with no $P_9$-minor. A 3-connected binary matroid $M$ has no $P_9$-minor if and only if $M$ is one of the internally 4-connected non-regular minors of a special…

Let AG(3,2)xU(1,1) denote the binary matroid obtained from the direct sum of AG(3,2) and a coloop by completing the 3-point lines between every element in AG(3,2) and the coloop. We prove that every internally 4-connected binary matroid…

The Splitter Theorem states that, if $N$ is a 3-connected proper minor of a 3-connected matroid $M$ such that, if $N$ is a wheel or whirl then $M$ has no larger wheel or whirl, respectively, then there is a sequence $M_0,..., M_n$ of…

Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cyclic…