Related papers: Matroids denser than a projective geometry
Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field $\mathbb F$, the list…
For a matroid of rank $r$ and a non-negative integer $k$, an element is called $k$-loose if every circuit containing it has size greater than $r-k$. Zaslavsky and the author characterized all binary matroids with a $1$-loose element. In…
We prove that for each prime power $q$ there is an integer $n$ such that if $M$ is a $3$-connected, representable matroid with a PG$(n-1,q)$-minor and no $U_{2,q^2+1}$-minor, then $M$ is representable over GF$(q)$. We also show that for…
In this sequel to "Foundations of matroids - Part 1", we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that the foundation of a matroid M is the colimit of the…
We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique…
We show that, if $M$ is a simple rank-$n$ matroid with no $\ell$-point line minor and no minor isomorphic to the cycle matroid of a $t$-vertex complete graph, then the ratio $|M| / n$ is bounded above by a singly exponential function of…
We show that a simple rank-$r$ matroid with no $(t+1)$-element independent flat has at least as many elements as the matroid $M_{r,t}$ defined as the direct sum of $t$ binary projective geometries whose ranks pairwise differ by at most $1$.…
The {\em Dressian} of a matroid $M$ is the set of all valuations of $M$. This Dressian is the support of a polyhedral complex $\mathcal{Dr}(M)$ whose open cells correspond 1-1 with matroid subdivisions of the matroid polytope of $M$. We…
Subject to hypotheses based on the matroid structure theory of Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of…
Let $M$ be a matroid. We study the expansions of $M$ mainly to see how the combinatorial properties of $M$ and its expansions are related to each other. It is shown that $M$ is a graphic, binary or a transversal matroid if and only if an…
A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. Positroid polytopes are matroid polytopes of positroids. In particular, they are lattice polytopes. The Ehrhart polynomial of a lattice polytope…
The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and…
We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a…
A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power…
We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating…
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 prove a conjecture of Geelen, Gerards, and Whittle that for any finite field $GF(q)$ and any integer $t$, every cosimple $GF(q)$-representable matroid with sufficiently large girth contains either $M(K_t)$ or $M(K_t)^*$ as a minor.
A matroid base polytope is a polytope in which each vertex has 0,1 coordinates and each edge is parallel to a difference of two coordinate vectors. Matroid base polytopes are described combinatorially by integral submodular functions on a…
We provide a formula for the Ehrhart polynomial of the connected matroid of size $n$ and rank $k$ with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $h^*$-real-rooted (and…
We call a class $\mathcal{M}$ of matroids hereditary if it is closed under flats. We denote by $\mathcal{M}^{ext}$ the class of matroids $M$ that is in $\mathcal{M}$, or has an element $e$ such that $M \backslash e$ is in $\mathcal{M}$. We…