Related papers: Excluding a line from $\mathbb C$-representable ma…

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…

For $n$ at least 7 and $n$ equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on $n$ strands. These generating sets are of the smallest possible cardinality. For $n$ equal to 4 or 6, we give…

For every positive integer $d$, we show that there must exist an absolute constant $c > 0$ such that the following holds: for any integer $n \geq cd^{7}$ and any red-blue coloring of the one-dimensional subspaces of $\mathbb{F}_{2}^{n}$,…

Let K be an arbitrary (commutative) field with at least three elements, and let n, p and r be positive integers with r<=min(n,p). In a recent work, we have proved that an affine subspace of M_{n,p}(K) containing only matrices of rank…

Let $k \ge 2$ be an integer. We show that if $s = 2$ and $t \ge 2$, or $s = t = 3$, then the maximum possible number of edges in a $C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$ is asymptotically equal to $(t - s +…

We show that for any regular matroid on $m$ elements and any $\alpha \geq 1$, the number of $\alpha$-minimum circuits, or circuits whose size is at most an $\alpha$-multiple of the minimum size of a circuit in the matroid is bounded by…

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…

An integer-valued matrix $\mathbf{A}$ is $\Delta$-modular if each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix has determinant at most $\Delta$ in absolute value. The column number problem is to determine the maximum…

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 $…

In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent…

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the…

We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular…

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…

We call a matroid element "loose" if it is contained in no circuits of size less than the rank of the matroid. A matroid in which all elements are loose is a paving matroid. Acketa determined all binary paving matroids, while Oxley…

We study the representation growth of alternating and symmetric groups in positive characteristic and restricted representation growth for the finite groups of Lie type. We show that the the number of representations of dimension at most n…

Let $d\ge4$ and $c\in(-d,d)$ be relatively prime integers. We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\le d\le 36$), the smallest prime $p\equiv c\pmod d$ with $p\ge(2dn-c)/(d-1)$ is the…

It is possible to write the indicator function of any matroid polytope as an integer combination of indicator functions of Schubert matroid polytopes. In this way, every matroid on $n$ elements of rank $r$ can be thought of as a lattice…

We prove that the number of single element extensions of $M(K_{n+1})$ is $2^{{n\choose n/2}(1+o(1))}$. This is done using a characterization of extensions as "linear subclasses".

Given positive integers $r$ and $c$, let $\phi(r,c)$ denote the number of isomorphism classes of complex rank $r$ topological vector bundles on $\mathbb{CP}^{r+c}$ that are stably trivial. We compute the $p$-adic valuation of the number…

Let r >= s >= 0 be integers and G be an r-graph. The higher inclusion matrix M_s^r(G) is a {0,1}-matrix with rows indexed by the edges of G and columns indexed by the subsets of V(G) of size s: the entry corresponding to an edge e and a…