In this paper we highlight some enumerative results concerning matroids of low rank and prove the tail-ends of various sequences involving the number of matroids on a finite set to be log-convex. We give a recursion for a new, slightly improved, lower bound on the number of rank-$r$ matroids on $n$ elements when $n=2^m-1$. We also prove an adjacent result showing the point-lines-planes conjecture to be true if and only if it is true for a special subcollection of matroids. Two new tables are also presented, giving the number of paving matroids on at most eight elements.
We introduce a connectivity function for infinite matroids with properties similar to the connectivity function of a finite matroid, such as submodularity and invariance under duality. As an application we use it to extend Tutte's linking theorem to finitary and to co-finitary matroids.
The notion of thin sums matroids was invented to extend the notion of representability to non-finitary matroids. A matroid is tame if every circuit-cocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a finite field k if and only if all its finite minors are representable over k. We expect that the method we use to prove this will make it possible to lift many theorems about finite matroids representable over a finite field to theorems about tame thin sums matroids over these fields. We give three examples of this: various characterisations of binary tame …
We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved recently by Aharoni and Berger. We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids. In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.
In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparse-paving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the $r$-subsets of $S$, $\binom{S}{r}=\sqcup_{i=1}^{γ}\mathcal{U}_{i}$ such that each $\mathcal{U}_{i}$ defines a sparse-paving matroid of rank $r$.
As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures. This new point of view also allows us to extend, and…
We present a new type of equivalence for representable matroids that uses the automorphisms of the underlying matroid. Two $r\times n$ matrices $A$ and $A'$ representing the same matroid $M$ over a field $F$ are {\it geometrically equivalent representations} of $M$ if one can be obtained from the other by elementary row operations, column scaling, and column permutations. Using geometric equivalence, we give a method for exhaustively generating non-isomorphic matroids representable over a finite field $GF(q)$, where $q$ is a power of a prime.
An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graph-like topological space: that is, a graph-like space in the sense of Thomassen and Vella. This extends Tutte's characterization of finite graphic matroids. The representation we construct has many pleasant topological properties. Working in the representing space, we prove that any circuit in a 3-connected graphic matroid is countable.
Let $G$ be a finite undirected multigraph with no self-loops. The Jacobian $\operatorname{Jac}(G)$ is a finite abelian group associated with $G$ whose cardinality is equal to the number of spanning trees of $G$. There are only a finite number of biconnected graphs $G$ such that the exponent of $\operatorname{Jac}(G)$ equals $2$ or $3$. The definition of a Jacobian can also be extended to regular matroids as a generalization of graphs. We prove that there are finitely many connected regular matroids $M$ such that $\operatorname{Jac}(M)$ has exponent $2$ and characterize all such matroids.
We consider the problem of determining whether the union of two infinite matroids is a matroid. We introduce a superclass of the finitary matroids, the nearly finitary matroids, and prove that the union of two nearly finitary matroids is a nearly finitary matroid. On the other hand, we prove that the union of two arbitrary infinite matroids is not necessarily a matroid. Indeed, we show (under a weak additional assumption) that the nearly finitary matroids are essentially the largest class of matroids for which one can have a union theorem. We then extend the base packing theorem for finite mat…
The main result is Theorem MAT 11 which states that every finite closure operator is the ground set of a matroid. Its base sets consist of nonredundant covers of of the closure. These are minimal subsets that determine the closure operator using a closure algorithm from the theory of relational databases. For each hereditary collection there is one flat closure that define a matroid or simplicial complexes, but every closure defines a hereditary collection, its keys in database theory. The normalization algorithm by Maier is the basis of this paper. This follows his development, and his releva…
It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals. In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to …
This paper investigates infinite matroids from a categorical perspective. We prove that the category of infinite matroids is a proto-exact category in the sense of Dyckerhoff and Kapranov, thereby generalizing our previous result on the category of finite matroids. We also characterize finitary matroids as co-limits of finite matroids, and show that the finite matroids are precisely the finitely presentable objects in this category.
A Note on an Analytic Approach to the Problem of Finite Matroid Representability, The Cardinality of Sets of k-Independent Vectors over Finite Fields and the Maximum Distance Separable Conjecture
Department of Mathematics, FIZ Karlsruhe Leibnitz Institute for Information Infrastructure（FIZ Karlsruhe莱布尼茨信息基础设施研究所数学系）；Department of Economics, University of Toronto（多伦多大学经济系）
The matroid-based valuation conjecture of Ostrovsky and Paes Leme states that all gross substitutes valuations on $n$ items can be produced from merging and endowments of weighted ranks of matroids defined on at most $m(n)$ items. We show that if $m(n) = n$, then this statement holds for $n \leq 3$ and fails for all $n \geq 4$. In particular, the set of gross substitutes valuations on $n \geq 4$ items is strictly larger than the set of matroid based valuations defined on the ground set $[n]$. Our proof uses matroid theory and discrete convex analysis to explicitly construct a large family of c…
We provide a weaker version of the generalized lifting property which holds in complete generality for all finite Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.
It is known that there are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating to the language of $h$-vectors, there are finitely many simplicial complexes of bounded dimension with $h_1=k$ for any natural number $k$. In this paper we study the question at the other end of the $h$-vector: Are there only finitely many $(d-1)$-dimensional simplicial complexes with $h_d=k$ for any given $k$? The answer is no if we consider general complexes, but when focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes…
One of the simplest axiomatizations of a matroid is in terms of independent sets. Curiously, no such independent set axiomatization is known for Gelfand and Serganova's WP-matroids (which are ``Coxeter group analogues'' of matroids). Here we state and prove such an axiomatization in the special case of symplectic matroids. No prior knowledge of WP-matroids is assumed; we believe the paper should be accessible and interesting to anyone with some interest in matroids.
Let $\cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $\cX$-matroid if each set in $\cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $\cX$-matroid in the weak order poset of all $\cX$-matroids on $E$, and characterizing its rank function when it exists.
To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid.
A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called "balanced", such that no theta subgraph contains exactly two balanced circles. A biased graph has two natural matroids, the frame matroid and the lift matroid. A classical question in matroid theory is whether a matroid can be embedded in a projective geometry. There is no known general answer, but for matroids of biased graphs it is possible to give algebraic criteria. Zaslavsky has previously given such criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces; in this pape…
The $\mathit{growth\ rate\ function}$ for a nonempty minor-closed class of matroids $\mathcal{M}$ is the function $h_{\mathcal{M}}(n)$ whose value at an integer $n \ge 0$ is defined to be the maximum number of elements in a simple matroid in $\mathcal{M}$ of rank at most $n$. Geelen, Kabell, Kung and Whittle showed that, whenever $h_{\mathcal{M}}(2)$ is finite, the function $h_{\mathcal{M}}$ grows linearly, quadratically or exponentially in $n$ (with base equal to a prime power $q$), up to a constant factor. We prove that in the exponential case, there are nonnegative integers $k$ and $d \le \…
We study the number of hamiltonian circuits, containing a fixed basis, and the number of hyperplanes, which do not contain a fixed basis in perfect matroid designs. Projective and affine finite geometries are considered as examples of such matroids. We give algorithms to find the hyperplanes and the hamiltonian circuits in such cases.
Motivated by Gröbner basis theory for finite point configurations, we define and study the class of "standard complexes" associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid duality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics.
A simple binary matroid, viewed as a restriction of a finite binary projective geometry $PG(n-1,2)$, is $I_{1,t}$-free if for any rank-$t$ flat of $PG(n-1,2)$, its intersection with the matroid is not a one-element set. In this paper, we show that the simple $I_{1,t}$-free and triangle-free binary matroids have bounded critical number for any $t \geq 1$.
Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of Boolean representable simplicial complexes. This turns out to be the direct sum of a complete matroid with a lift matroid of the complete biased graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar biased graph. This gives a new perspective on both matroids and also an application of matroid theory to the theory of fini…
Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits, and the edges of T correspond to certain nested 2-separations of M. These decompositions are invariant under duality.
New families of matroids are constructed in this note. These new fami- lies are derived from the concept of linearly independent set family (LISF) introduced by Eicker and Ewald [Linear Algebra and its Applications 388 (2004) 173-191]. The proposed construction generalizes in a natural way the well known class of vectorial matroids over a field.
A matroid is Ingleton if all quadruples of subsets of its ground set satisfy Ingleton's inequality. In particular, representable matroids are Ingleton. We show that the number of Ingleton matroids on ground set $[n]$ is doubly exponential in $n$; it follows that almost all Ingleton matroids are non-representable.
We study the relationship between a q-analogue of matroids and linear codes with the rank metric in the vector space of matrices with entries in a finite field. We prove a Greene type identity for the rank generating function of these matroidal structures and the rank weight enumerator of these linear codes. As an application, we give a combinatorial proof of a MacWilliams type identity for Delsarte rank-metric codes.
We prove that for each finite field $\mathbb F$ and integer $k\in \mathbb Z$ there exists $n\in \mathbb Z$ such that no excluded minor for the class of $\mathbb F$-representable matroids has $n$ nested $k$-separations.
This is a survey article on the exchange properties characterizing M-natural-concave set functions and valuated matroids (M-concave set functions). The objective of this paper is to collect related results scattered in the literature and to give (reasonably) self-contained elementary proofs for them.
We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gröbner fan of the defining ideal of our Gorenstein algebra.
Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph.
We introduce the active partition of the ground set of an oriented matroid perspective (or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the …
We show that, for each real number $α> 0$ and odd integer $k\ge 5$ there is an integer $c$ such that, if $M$ is a simple binary matroid with $|M| \ge α2^{r(M)}$ and with no $k$-element circuit, then $M$ has critical number at most $c$. The result is an easy application of a regularity lemma for finite abelian groups due to Green.
Let $M$ be a representable matroid, and $Q, R, S, T$ subsets of the ground set. We prove that, if $M$ is sufficiently large, then there is an element $e$ such that deleting or contracting $e$ preserves both the $Q$-$R$ and the $S$-$T$ connectivities. For matroids representable over a finite field we prove a stronger result: we show that we can remove $e$ such that both a connectivity and a minor of $M$ are preserved.
Let $M_1,M_2,\ldots,M_k$ be a collection of matroids on the same ground set $E$. A coloring $c:E \rightarrow \{1,2,\ldots,k\}$ is called \emph{cooperative} if for every color $j$, the set of elements in color $j$ is independent in $M_j$. We prove that such coloring always exists provided that every matroid $M_j$ is itself $k$-colorable (the set $E$ can be split into at most $k$ independent sets of $M_j$). We derive this fact from a generalization of Seymour's list coloring theorem for matroids, which asserts that every $k$-colorable matroid is $k$-list colorable, too. We also point on some con…
Let $ E $ be a possibly infinite set and let $ M $ and $ N $ be matroids defined on $ E $. We say that the pair $ \{ M,N \} $ has the Intersection property if $ M $ and $ N $ share an independent set $ I $ admitting a bipartition $ I_M\sqcup I_N $ such that $ \mathsf{span}_M(I_M)\cup \mathsf{span}_N(I_N)=E $. The Matroid Intersection Conjecture of Nash-Williams says that every matroid pair has the Intersection property. The conjecture is known and easy to prove in the case when one of the matroids is uniform and it was shown by Bowler and Carmesin that the conjecture is implied by its special …
Let $M$ and $N$ be two matroids on the same ground set. We generalize results of Drisko and Chapell by showing that any $2n-1$ sets of size $n$ in $M \cap N$ have a rainbow set of size $n$ in $M \cap N$.
We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications. These include a new approach to cluster analysis and image segmentation. As another illustration for the abstract theorem, we show that applying it to edge-tangles yields the Gomory-Hu theorem.
We give an exposition of some results from matroid theory which characterise the finite pregeometries arising from Hrushovski's predimension construction. As a corollary, we observe that a finite pregeometry which satisfies Hrushovski's flatness condition arises from a predimension.
A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k>4) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the …
The ground set for all matroids in this paper is the set of all edges of a complete graph. The notion of a {\it maximum matroid for a graph} $G$ is introduced, and the existence and uniqueness of the maximum matroid for any graph $G$ is proved. The maximum matroid for $K_3$ is shown to be the cycle (or graphic) matroid. This result is pursued in two directions - to determine the maximum matroid for the $m$-cycle $C_m$ and to determine the maximum matroid for the complete graph $K_m$. The maximum matroid for $K_4$ is the matroid whose bases are the Laman graphs, related to structural rigidity o…
We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $r^c$ rank-$a$ sets, or $\mathcal{M}$ contains the GF$(q)$-representable matroids for some prime power $q$ and every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $cq^r$ rank-$a$ sets. In the latter case, this determines the maximum density of matroids in $\mathcal{M}$ up to a constant factor.
