We prove that a biased graph is gainable over the group 𝖹_3 if and only if it contains no minor isomorphic to (4K_2,∅), ± K_3, or -K_4. We develop a theory of "partial groups" that is analogous to that of partial fields, and we use this theory to show that a biased graph is gainable over every non-trivial group if and only if it is gainable over 𝖹_2 and 𝖹_3. From this we derive an independent proof of the theorem due to Gerards that a biased graph is gainable over every non-trivial group if and only if it has no minor isomorphic to (3K_2,∅), ± K_3, or -K_4.

We present a Myhill-Nerode theorem for hypergraphs. The theorem involves an operation which takes two input structures and produces a hypergraph as output. Using this operation, we define a Myhill-Nerode-type equivalence relation and show that if a class of hypergraphs is definable in the counting monadic second-order logic of hypergraphs, then the equivalence relation has finite index. We apply this tool to classes of gain-graphic matroids, and show that if the group Γ is not uniformly locally finite, then the class of Γgain-graphic matroids is not monadically definable. (A group is uniformly locally finite if, for every k, there is a maximum size amongst subgroups generated by at most k elements.) In addition, we define the conviviality graph of a group, and show that if the group Γ has an infinite conviviality graph, then the class of Γgain-graphic matroids is not monadically definable. This will be useful in future constructions.

A matroid is supersolvable if it has a maximal chain of flats, each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and we show that several results for chordal graphs hold in this matroidal context. In particular, we consider matroid analogues of the reduced clique graph and clique trees for chordal graphs. The latter is a maximum-weight spanning tree of the former. We also show that the matroid analogue of a clique tree is an optimal decomposition for the matroid parameter of tree-width.

We review the work of Dominic Welsh (1938-2023), tracing his remarkable influence through his theorems, expository writing, students, and interactions. He was particularly adept at bringing different fields together and fostering the development of mathematics and mathematicians. His contributions ranged widely across discrete mathematics over four main career phases: discrete probability, matroids and graphs, computational complexity, and Tutte-Whitney polynomials. We give particular emphasis to his work in matroid theory and Tutte-Whitney polynomials.

A transduction provides us with a way of using the monadic second-order language of a structure to make statements about a derived structure. Any transduction induces a relation on the set of these structures. This article presents a self-contained presentation of the theory of transductions for the monadic second-order language of matroids. This includes a proof of the matroid version of the Backwards Translation Theorem, which lifts any formula applied to the images of the transduction into a formula which we can apply to the pre-images. Applications include proofs that the class of lattice-path matroids and the class of spike-minors can be defined by sentences in monadic second-order logic.

Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph G. The nodes of the reduced clique graph are the maximal cliques of G, and two nodes are joined by an edge if and only if they form a non-disjoint separating pair of cliques in G. In this case the weight of the edge is the size of the intersection of the two cliques. A clique tree of G is a tree with the maximal cliques of G as its nodes, where for any v∈ V(G) , the subgraph induced by the nodes containing v is connected. Galinier et al. prove that a spanning tree of the reduced clique graph is a clique tree if and only if it has maximum weight, but their proof contains an error. We explain and correct this error. In addition, we initiate a study of the structure of reduced clique graphs by proving that they cannot contain any induced cycle of length five (although they may contain induced cycles of length three or any even integer greater than two). We show that no cycle of length four or more is isomorphic to a reduced clique graph. We prove that the class of clique graphs of chordal graphs is not comparable to the class of reduced clique graphs of chordal graphs by providing examples that are in each of these classes without being in the other.

We conjecture that the class of frame matroids can be characterised by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterisation for the class of bicircular matroids. The proof does not depend on an excluded-minor characterisation.

Hlineny's Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of F-representable matroids with bounded branch-width (where F is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the class is pigeonhole. An efficiently pigeonhole class has a stronger property, involving an efficiently-computable equivalence relation on subsets of the ground set. We show that Hlineny's Theorem extends to any efficiently pigeonhole class. In a sequel paper, we use these ideas to extend Hlineny's Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, where H is any finite group. We also give a characterisation of the families of hypergraphs that can be described via tree automata: a family is defined by a tree automaton if and only if it has bounded decomposition-width. Furthermore, we show that if a class of matroids has the pigeonhole property, and can be defined in monadic second-order logic, then any subclass with bounded branch-width has a decidable monadic second-order theory.

This dataset contains the 385370 pairwise non-isomorphic matroids from 0 to 9 elements inclusive. Each matroid is given as a text string occupying one line of the file matroids09_rankLine. Except for the empty matroid, the text string gives the rank of every subset of the groundset (as described in more detail in the readme.md file), and so is called the rankline of the matroid. This establishes a de-facto numbering scheme for the matroids, starting with M0 (the empty matroid) and ending with M385369 (the uniform matroid U(9,9)). For example, M298 is the number for the Fano plane, and its rankLine occurs on line 298 (counting from 0) of the file. Various published papers in matroid theory have already referred to specific matroids using the M-numbers obtained from this catalogue. This dataset is provided as a "reference version" of the catalogue to permit researchers, both now and into the future, to safely use M-numbers as unique and unchanging identifiers to refer to any matroid on up to 9 elements.

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 flats. We confirm a conjecture of Joswig and Schröter by proving an excluded-minor characterisation of the class of split matroids.

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 number of 2t-element excluded minors. However, there are only finitely many excluded minors with ground sets of odd size.

DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid Un;2n or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.

A connected matroid $M$ is unbreakable if, for each of its flats $F$, the matroid $M/F$ is connected or, equivalently, if $M^*$ has no two skew circuits. Pfeil showed that a simple graphic matroid $M(G)$ is unbreakable exactly when $G$ is either a cycle or a complete graph. We extend this result to describe which graphs are the underlying graphs of unbreakable frame matroids.

Let $M$ be a representable matroid on $n$ elements. We give bounds, in terms of $n$, on the least positive characteristic and smallest field over which $M$ is representable.

A pair $(A,B)$ of square $(0,1)$-matrices is called a Lehman pair if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called Lehman matrices. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to partitionable graphs (also known as $(\alpha,\omega)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular $6$-vertex subgraph that we call a $3$-rung ladder segment. Two decades ago, Lütolf & Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue (which has just one omission), and extend the computational results to $20 \times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order up to $20 \times 20$, only two do not arise from our $3$-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with $k=1$.

We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case, these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transversals, and linear codes over division rings, in each case yielding a duality theorem for the respective class of objects.

Let M be a representable matroid on n elements. We give bounds, in terms of n, on the least positive characteristic and smallest field over which M is representable. Our starting point is given by the following two theorems of Rado [5]. Theorem 1 (Rado, 1957). Let M be a matroid representable over a field K. Then M is representable over a simple algebraic extension of the prime field of K. Theorem 2 (Rado, 1957). Let K be an extension field of Q of degree N , and let M be a matroid representable over K. Then there is a positive integer c such that given any prime p > c there is a positive integer k = k(p) ≤ N such that M is representable over GF(pk). For infinitely many p, k(p) = 1. Together, these two theorems say that if a matroid is linearly representable, then it is representable over a finite field. We ask, given a representable matroid on n elements, how large must such a field be? That is, given an n-element representable matroid M , what bound, depending just on n, can we place on the size of a field required to represent M? To that end, let Mn be the set of all representable matroids on n elements. For a matroid M , let c(M) be the least positive characteristic of a field over which M is representable. For each positive integer n, define c(n) = max{c(M) : M ∈ Mn}. Department of Pure Mathematics, University of Waterloo, Canada. jpbell@uwaterloo.ca School of Mathematics and Statistics, Victoria University of Wellington, New Zealand. Current affiliation: Dept. of Mathematics, Douglas College, Canada. funkd@douglascollege.ca School of Mathematics and Statistics, Victoria University of Wellington, New Zealand. byoungdu.kim@vuw.ac.nz, dillon.mayhew@vuw.ac.nz

We prove there is no sentence in the monadic second-order language MS0 that characterises when a matroid is representable over at least one field, and no sentence that characterises when a matroid is K-representable, for any infinite field K. By way of contrast, because Rota's Conjecture is true, there is a sentence that characterises F-representable matroids, for any finite field F.

Let M be a representable matroid on n elements. We give bounds, in terms of n, on the least positive characteristic and smallest field over which M is representable. Our starting point is given by the following two theorems of Rado [5]. Theorem 1 (Rado, 1957). Let M be a matroid representable over a field K. Then M is representable over a simple algebraic extension of the prime field of K. Theorem 2 (Rado, 1957). Let K be an extension field of Q of degree N , and let M be a matroid representable over K. Then there is a positive integer c such that given any prime p > c there is a positive integer k = k(p) ≤ N such that M is representable over GF (pk). For infinitely many p, k(p) = 1. Together, these two theorems say that if a matroid is linearly representable, then it is representable over a finite field. We ask, given a representable matroid on n elements, how large must such a field be? That is, given an n-element representable matroid M , what bound, depending just on n, can we place on the size of a field required to represent M? To that end, let Mn be the set of all representable matroids on n elements. For a matroid M , let c(M) be the least positive characteristic of a field over which M is representable. For each positive integer n, define c(n) = max{c(M) : M ∈ Mn}. Let f(M) be the order of the smallest field over which M is representable. For each positive integer n, define f(n) = max{f(M) : M ∈ Mn}. By Rado’s Theorems 1 and 2 above, c(n) exists and f(n) is finite for all n. Note that c(n) ≤ f(n) for all n, and that, since adding a loop to an n-element matroid yields a matroid on n+1 elements representable over exactly the same fields, c and f are non-decreasing. A result of Brylawski [1] provides a lower bound for c (and thus for f ; see Section 4). We ask for upper bounds on c(n) and f(n). For matroids on at most 8 elements, Table 1 summarises the data (the fact that f(8) = 11 is courtesy G. Royle [personal communication]. We obtain the following bounds. Date: March 2, 2018. Supported by a Rutherford Discovery Fellowship. 1 2 BELL, FUNK, KIM, AND MAYHEW n c(n) f(n) 1 2 2 2 2 2 3 2 2 4 2 3 5 2 4 6 2 5 7 3 7 8 ? 11 Table 1 Theorem 3. For all positive integers n, log2 log2 c(n) ≤ n and log2 log2 log2 f(n) ≤ n. The following fact falls out of the proof of Theorem 3. Theorem 4. Let M be an n-element matroid representable over a field of characteristic 0, and let p be a prime satisfying log2 log2 log2 p > n . Then M is representable over GF(p). We consider the cases of representability over only positive characteristic (Theorem 2.1) and representability over characteristic 0 (Theorem 3.1) separately. Theorem 3 then follows immediately from these results. By Table 1, we may assume throughout the rest of the paper that n > 7. 1. Bounding the degree of a field extension Our first step is to prove an effective version of Rado’s Theorem 1: Theorem 1.1. Let M be a matroid on n elements representable over a field K. Then M is representable over a simple algebraic extension of the prime field of K of degree at most 22 2n 2 . 1.1. A system of polynomials arising from a matroid. Our approach is a standard one in studies of representability of matroids over fields. We assign to an n-element, rank-r matroid M an r × n matrix A whose entries are indeterminates x1, . . . , xt, where t = rn. Each element of the matroid is represented by a column of the matrix. From this matrix we obtain a system of polynomial equations in Z[x1, . . . , xt] as follows. For each r-element subset X of the ground set of M , there is a corresponding r× r submatrix of A whose columns are those representing the elements in X. Setting the determinants of r × r submatrices corresponding to dependent sets to zero, and demanding that the determinants of those r × r submatrices that correspond to bases be nonzero, yields a system of polynomials. The latter conditions may be expressed by multiplying each polynomial fi obtained