A Theorem on Matroid Homomorphism

A Cantor-Bernstein theorem for infinite matroids

We give a common matroidal generalisation of `A Cantor-Bernstein theorem for paths in graphs' by Diestel and Thomassen and `A Cantor-Bernstein-type theorem for spanning trees in infinite graphs' by ourselves.

Towards a splitter theorem for internally 4-connected binary matroids VIII: small matroids

Our splitter theorem for internally 4-connected binary matroids studies pairs of the form (M,N), where N and M are internally 4-connected binary matroids, M has a proper N-minor, and if M' is an internally 4-connected matroid such that M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The analysis in the splitter theorem requires the constraint that |E(M)|>15. In this article, we complement that analysis by using an exhaustive computer search to find all such pairs satisfying |E(M)|<16.

An obstacle to a decomposition theorem for near-regular matroids

Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obtained from matroids that belong to a few basic classes through k-sums. Also suppose that these basic classes are such that, whenever a class contains all graphic matroids, it does not contain all cographi…

Unifying duality theorems for width parameters in graphs and matroids. II. General duality

We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]

Logarithmic concavity for morphisms of matroids

Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of basis for morphisms of matroids, and show that its generating function is strongly log-concave. As a consequence, we obtain a generalization of Mason's conjecture on the $f$-vectors of independent subsets of matroids to arbitrary morphisms of matroids. To establish this, we define multivariate Tutte polynomials of morphisms of matroids, and show that they are Lorentzian in the sense of [BH19] for sufficiently small positive parameters.

A Topological Representation Theorem for Tropical Oriented Matroids: Part II

Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes -- in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper we show that this correspondence is a bijection. Moreover, a tropical analogue for the Topol…

An Algorithmic Proof of the Piff--Welsh Theorem on Transversal Matroid Representations

A fundamental theorem of matroid theory establishes that a transversal matroid is representable over fields of any characteristic. It was proved in 1970 by Piff and Welsh: their proof is elegant and concise and, moveover, constructive. However it is far from being algorithmic, in terms of suggesting a step-by-step procedure for deriving a collection of vectors over a given base field representing the transversal matroid induced by a given set system. In this note we recast Piff and Welsh's proof in algorithmic form.

关于transversal $q$-matroids的循环平坦嵌入定理

A cyclic flat embedding theorem for transversal $q$-matroids

本文通过循环平坦结构建立matroid与transversal $q$-matroid的对应关系，证明了transversal matroid的循环平坦结构在对应下保持不变，并展示了nested $q$-matroid的可表示性。

A two-dimensional topological representation theorem for matroid polytopes of rank 4

The Folkman-Lawrence topological representation theorem, which states that every (loop-free) oriented matroid of rank $r$ can be represented as a pseudosphere arrangement on the $(r-1)$-dimensional sphere $S^{r-1}$, is one of the most outstanding results in oriented matroid theory. In this paper, we provide a lower-dimensional version of the topological representation theorem for uniform matroid polytopes of rank $4$. We introduce $2$-weak configurations of points and pseudocircles ($2$-weak PPC configurations) on $S^2$ and prove that every uniform matroid polytope of rank $4$ can be represent…

KKM定理与超越拟阵的离散几何

KKM theorems and discrete geometry beyond matroids

引入选择结构，一种拓扑框架，扩展了颜色类和拟阵在离散几何与KKM定理中的作用，将经典结果推广到非拟阵例子（如棋盘复和匹配复），并推广了多个拟阵版本的经典定理。

A Density Hales-Jewett Theorem for matroids

We show that, if $α> 0$ is a real number, $n \ge 2$ and $\ell \ge 2$ are integers, and $q$ is a prime power, then every simple matroid $M$ of sufficiently large rank, with no $U_{2,\ell}$-minor, no rank-$n$ projective geometry minor over a larger field than $\GF(q)$, and satisfying $|M| \ge αq^{r(M)}$, has a rank-$n$ affine geometry restriction over $\GF(q)$. This result can be viewed as an analogue of the Multidimensional Density Hales-Jewett Theorem for matroids.

可测拟阵：基础与极小-极大定理

Measurable Matroids: Foundations and Min--Max Theorems

本文在标准无原子测度空间建立可测拟阵理论，证明其核心定理并给出图论匹配、分解等应用，拓展了经典拟阵理论的测度类比框架。

Towards a splitter theorem for internally 4-connected binary matroids II

Let M and N be internally 4-connected binary matroids such that M has a proper N-minor, and |E(N)| is at least seven. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M\e has no N-minor whenever e is in a triangle of M, and M/e has no N-minor whenever e is in a triad of M, then M has a minor, M', such that M' is internally 4-connected with an N-minor, and 0 < |E(M)|-|E(M')| < 3.

A Tverberg type theorem for matroids

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets σ_1,\ldots,σ_t \in M such that \bigcap_{i=1}^t f(σ_i) \neq \emptyset.

KKM覆盖的拟阵着色

Matroid colorings of KKM covers

本研究提出并证明了多面体集合覆盖的拟阵着色族的KKM型定理，该定理推广了Gale的彩色KKM定理及相关近期稀疏彩色变体结果。

On the Quotient-Lift Matroid Relation

It is well known that a matroid L is a lift of a matroid M if and only if every circuit of L is the union of some circuits of M. In this paper we give a simpler proof of this important theorem. We also described a discrete homotopy theorem on two matroids of different ranks on the same ground set.

The structure of delta-matroids with width one twists

The width of a delta-matroid is the difference in size between a maximal and minimal feasible set. We give a Rough Structure Theorem for delta-matroids that admit a twist of width one. We apply this theorem to give an excluded minor characterisation of delta-matroids that admit a twist of width at most one.

Applications of the Strong Splitter Theorem: decomposition results

We use the Strong Splitter Theorem to decompose the excluded minor class of binary matroids with no $E_4$-minor. Using this theorem we can get the 3-decomposers and the extremal internally 4-connected matroids as well as any other important matroids in the class. The matroid $E_4$ is a self-dual 10-element binary 3-connected matroid that plays a useful role in structural results. It is a single-element coextension of $P_9$, which is a single-element extension of the 4-wheel. We show that the extremal matroids in this class are the binary rank-$r$ spikes $Z_r$, the rank 3 and 4 projective geome…

Tropical Carathéodory with Matroids

Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the tropical colorful Carathéodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. In particular, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programm…

Tangle-tree duality: in graphs, matroids and beyond

We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets. Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a duality theorem for tree-width. Our results can be used to derive short proofs of all the classical duality theorems for wi…

Countable Menger theorem with finitary matroid constraints on the ingoing edges

We present a strengthening of the countable Menger theorem (edge version) of R. Aharoni. Let $ D=(V,A) $ be a countable digraph with $ s\neq t\in V $ and let $\mathcal{M}=\bigoplus_{v\in V}\mathcal{M}_v $ be a matroid on $ A $ where $ \mathcal{M}_v $ is a finitary matroid on the ingoing edges of $ v $. We show that there is a system of edge-disjoint $ s \rightarrow t $ paths $ \mathcal{P} $ such that the united edge set of the paths is $ \mathcal{M} $-independent, and there is a $ C \subseteq A $ consists of one edge from each element of $ \mathcal{P} $ for which $ \mathsf{span}_{\mathcal{M}}(…

Profiles of separations: in graphs, matroids and beyond

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.

Matroid intersection, base packing and base covering for infinite matroids

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…

Infinite matroid union

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…

Lift theorems for representations of matroids over pastures

Pastures are a class of field-like algebraic objects which include both partial fields hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results even in the case of lifts of partial fields, for example the conjecture of Pendavingh--van Zwam that their lift construction is idempotent. We give numerous applications to matroid representations…

Inductive tools for connected ribbon graphs, delta-matroids and multimatroids

We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.

拟阵拓扑zeta函数的主定理

A master theorem for topological zeta functions of matroids

本文通过提出一个主定理，完整描述了无环拟阵的拓扑zeta函数及其Möbius变换的所有系数，并解决了van der Veer (2019)、Kutler (2023)和Mengesha, Miranda, Sun (2026)的猜想。

形式化 Seymour 矩阵分解定理的蓝图

A Blueprint for the Formalization of Seymour's Matroid Decomposition Theorem

本文提出通过完全单模表示模块化地形式化正则矩阵体结构，证明正则性在合并操作下保持不变，并为图、余图和 R_{10} 矩阵体建立正则性。

复可表示拟阵的西尔维斯特-加莱型定理

A Sylvester-Gallai-type theorem for complex-representable matroids

本文证明了复可表示拟阵的西尔维斯特-加莱型定理，即秩至少为$4^{k-1}$的复可表示拟阵必含恰好$k$个点的秩-$k$平坦，且通过更初等的证明获得了更优界。

Some heterochromatic theorems for matroids

The anti-Ramsey number of Erdös, Simonovits and Sós from 1973 has become a classic invariant in Graph Theory. To study this invariant in Matroid Theory, we use a related invariant introduce by Arocha, Bracho and Neumann-Lara. The heterochromatic number $hc(H)$ of a non-empty hypergraph $H$ is the smallest integer $k$ such that for every colouring of the vertices of $H$ with exactly $k$ colours, there is a totally multicoloured hyperedge of $H$. Given a rank-$r$ matroid $M$, there are several hypergraphs associated to the matroid that we can consider. One is $C(M) $, the hypergraph where the po…

伪可定向带图：矩阵-准树定理和对数凹性

Pseudo-orientable ribbon graphs: Matrix--Quasi-tree Theorem and log-concavity

本文研究了伪可定向带图的性质，推导出矩阵-准树定理和对数凹性结果，并展示了非伪可定向带图的实例。

Splitters and Decomposers for Binary Matroids

Let $EX[M_1\dots, M_k]$ denote the class of binary matroids with no minors isomorphic to $M_1, \dots, M_k$. In this paper we give a decomposition theorem for $EX[S_{10}, S_{10}^*]$, where $S_{10}$ is a certain 10-element rank-4 matroid. As corollaries we obtain decomposition theorems for the classes obtained by excluding the Kuratowski graphs $EX[M(K_{3,3}), M^*(K_{3,3}), M(K_5), M^*(K_5)]$ and $EX[M(K_{3,3}), M^*(K_{3,3})]$. These decomposition theorems imply results on internally $4$-connected matroids by Zhou [\ref{Zhou2004}], Qin and Zhou [\ref{Qin2004}], and Mayhew, Royle and Whitte [\ref…

The Axiomatization of Affine Oriented Matroids Reassessed

In an unpublished manuscript of 1992, Johan Karlander has given an axiomatization of affine oriented matroids. A closer examination of the text revealed an invalid construction and an incorrect argument in the proof of his main theorem. This paper provides an alternative argument to fix and slightly simplify the proof of the main theorem.

A unique factorization theorem for matroids

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 function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids $M$ and $N$ is maximal with respect to the weak order among matroids having $M$ as a submatroid, with complementary contraction equal to $N$. Any minor of the free product of $M$ and $N$ is a free product of a repeated t…

树自动机与拟阵的鸽巢类：II

Tree automata and pigeonhole classes of matroids: II

本文应用前序论文结果，将基于分支宽度的固定参数易处理算法扩展至基本横截拟阵、格路拟阵、双环拟阵和$H$-增益图拟阵等高效鸽巢类，并由此给出Courcelle定理的新证明。

无限拟阵的环划分

Circuit-partition of infinite matroids

本文为Komjáth等人关于有限匹配可扩展拟阵环划分的紧致性定理提供了简短证明，并推广了Thomassen猜想，证明了有限二元有向拟阵划分为有向环的充要条件。

Strong Splitter Theorem

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 3-connected matroids with $M_0\cong N$, $M_n=M$ and for $i\in \{1,..., n\}$, $M_i$ is a single-element extension or coextension of $M_{i-1}$. Observe that there is no condition on how many extensions may occur before a coextension must occur. In this paper, we give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, $M$ …

The structure of $I_4$-free and triangle-free binary matroids

A simple binary matroid is called $I_4$-free if none of its rank-4 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $PG(n-1,2)$ for which $|E \cap F|$ is not a basis of $F$ for any four-dimensional flat $F$. We prove a decomposition theorem that exactly determines the structure of all $I_4$-free and triangle-free matroids. In particular, our theorem implies that the $I_4$-free and triangle-free matroids have critical number at most $2$.

$N$-detachable pairs in 3-connected matroids III: the theorem

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an $N$-minor. This is the third and final paper in a series where we prove that if $|E(M)|-|E(N)| \ge 10$, then either $M$ has an $N$-detachable pair after possibly performing a single $Δ$-$Y$ or $Y$-$Δ$ exchange, or $M$ is essentially $N$ with a spike attached. Moreover, we describe the additional structures that arise if we require only that $|E(M)|-|E(N)| \ge 5$.

Decomposition of Quaternary Signed-Graphic Matroids

In this work we provide a decomposition theorem for the class of quaternary and non-binary signed-graphic matroids. This generalizes previous results for binary signed-graphic matroids and graphic matroids, and it provides the theoretical basis for a recognition algorithm.

Oriented matroids and Ky Fan's theorem

L. Lovasz has shown that Sperner's combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. Inspired by this result we prove that classical Ky Fan's theorem admits an oriented matroid generalization of similar nature. Ky Fan's theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C^{m,r} .

Lifts of matroid representations over partial fields

There exist several theorems which state that when a matroid is representable over distinct fields F_1,...,F_k, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First, parts of Whittle's characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a matroid is representable over both GF(4) and GF(5), then it is representable over the real numbers by a matrix such that the absolute value of the determinant of every nonsingular square submatrix is a powe…

A splitter theorem on 3-connected matroids and graphs

We establish the following splitter theorem for graphs and its generalization for matroids: Let $G$ and $H$ be $3$-connected simple graphs such that $G$ has an $H$-minor and $k:=|V(G)|-|V(H)|\ge 2$. Let $n:=\left\lceil k/2\right\rceil+1$. Then there are pairwise disjoint sets $X_1,\dots,X_n\subseteq E(G)$ such that each $G/X_i$ is a $3$-connected graph with an $H$-minor, each $X_i$ is a singleton set or the edge set of a triangle of $G$ with $3$ degree-$3$ vertices and $X_1\cup\cdots\cup X_n$ contains no edge sets of circuits of $G$ other than the $X_i$'s. This result extends previous ones of …

有向拟阵中的彩色回路与彩色拓扑

Colorful circuits and colorful topes in oriented matroids

本文给出有向拟阵中彩色Carathéodory定理圆锥版本的简短证明，并建立关于拓扑的彩色定理，其方法基于Sperner引理与Meshulam引理的共同推广。

A Polyhedral Proof of the Matrix Tree Theorem

The classical matrix tree theorem relates the number of spanning trees of a connected graph with the product of the nonzero eigenvalues of its Laplacian matrix. The class of regular matroids generalizes that of graphical matroids, and a generalization of the matrix tree theorem holds for this wider class. We give a new, geometric proof of this fact by showing via a dissect-and-rearrange argument that two combinatorially distinct zonotopes associated to a regular matroid have the same volume. Along the way we prove that for a regular oriented matroid represented by a unimodular matrix, the latt…

On Seymour's Decomposition Theorem

Let $\mathcal M$ be a class of matroids closed under minors and isomorphism. Let $N$ be a matroid in $\mathcal M$ with an exact $k$-separation $(A, B)$. We say $N$ is a $k$-decomposer for $\mathcal M$ having $(A, B)$ as an inducer, if every matroid $M\in \mathcal M$ having $N$ as a minor has a $k$-separation $(X, Y)$ such that, $A\subseteq X$ and $B\subseteq Y$. Seymour [3, 9.1] proved that a matroid $N$ is a $k$-decomposer for an excluded-minor class, if certain conditions are met for all 3-connected matroids $M$ in the class, where $|E(M)-E(N)|\le 2$. We reinterpret Seymour's Theorem in term…

Tverberg type theorems for matroids

In this paper we show a variant of colorful Tverberg's theorem which is valid in any matroid: Let $S$ be a sequence of non-loops in a matroid $M$ of finite rank $m$ with closure operator cl. Suppose that $S$ is colored in such a way that the first color does not appear more than $r$-times and each other color appears at most $(r-1)$-times. Then $S$ can be partitioned into $r$ rainbow subsequences $S_1,\ldots, S_r$ such that $cl\,\emptyset\subsetneq cl\,S_1\subseteq cl\, S_2\subseteq \ldots \subseteq cl\,S_r$. In particular, $\emptyset\neq \bigcap_{i=1}^r cl\,S_i$. A subsequence is called rainb…

A decomposition theorem for binary matroids with no prism minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connectio…

Some Matroids Related to Sum-Rank Metric Codes

We introduce the notion of sum-matroids and show its association with sum-rank metric codes. As a consequence, some results for sum-rank metric codes by Martínez-Peñas are generalized for sum-matroids. The sum-matroids generalize the notions of matroids and $q$-matroids. We define the generalized weights for sum-matroids and prove a Wei-type duality theorem which generalizes the analogous cases for matroids and q-matroids.

Drisko定理关于两个拟阵交的加强的一个共同推广

A common generalization to strengthenings of Drisko's Theorem for intersections of two matroids

本文证明了在同一基集上的两个拟阵中，存在大小为n的部分彩虹独立集，共同推广了二部图和拟阵交的彩虹匹配结果。

关于\(\operatorname{GF}(q)\)可表示拟阵的柯瓦里 - 索斯 - 图兰定理

The Kővari-Sós-Turán theorem for $\operatorname{GF}(q)$-representable matroids

研究关于\(\operatorname{GF}(q)\)可表示拟阵的柯瓦里 - 索斯 - 图兰定理类似结果，通过特定条件得出\(|E(M)|\)的上界，尤其对无\(M(K_{2,t})\)限制的简单秩\(n\)二元拟阵确定元素最大数量及下界获取方式。

Finiteness theorems for matroid complexes with prescribed topology

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…

The structure of claw-free binary matroids

A simple binary matroid is called claw-free if none of its rank-3 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $\mathrm{PG}(n-1,2)$ for which $|E \cap P|$ is not a basis of $P$ for any plane $P$, or as the subsets $X$ of $\mathbb{F}_2^n$ containing no linearly independent triple $x,y,z$ for which $x+y,y+z,x+z,x+y+z \notin X$. We prove a decomposition theorem that exactly determines the structure of all claw-free matroids. The theorem states that claw-free matroids either belong to one of three particular basic classes of claw-free matroids,…

Typical structure of hereditary properties of binary matroids

We prove an arithmetic analogue of the typical structure theorem for graph hereditary properties due to Alon, Balogh, Bollobás and Morris.