Related papers: The joints problem for matroids

We investigate the maximum number of intersections between two polygons with p and q vertices, respectively, in the plane. The cases where p or q is even or the polygons do not have to be simple are quite easy and already known, but when p…

Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present…

We introduce the parametric matroid one-interdiction problem. Given a matroid, each element of its ground set is associated with a weight that depends linearly on a real parameter from a given parameter interval. The goal is to find, for…

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 give two proofs that the $h$-vector of any paving matroid is a pure O-sequence, thus answering in the affirmative a conjecture made by R. Stanley, for this particular class of matroids. We also investigate the problem of obtaining good…

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$.…

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 $1$.…

In this short note we apply a recent theorem of Koll\'ar about the arithmetic genus of curves to give a bound on the number of joints weighted by the multiplicities. This gives an affirmative answer to a conjecture of Carbery in the generic…

We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not…

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,…

Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$…

It is shown that if $F$ denotes the number of filled cells in a superimposed pair of maximal orthogonal partial Latin squares of order $n$, then $F\ge n^2/3$. This resolves a conjecture raised in an earlier paper by the current authors. It…

Let $M$ be a matroid without loops or coloops and let $T(M;x,y)$ be its Tutte polynomial. In 1999 Merino and Welsh conjectured that $$\max(T(M;2,0), T(M;0,2))\geq T(M;1,1)$$ holds for graphic matroids. Ten years later, Conde and Merino…

We consider the combinatorial question of how many convex polygons can be made by using the edges taken from a fixed triangulation of n vertices. For general triangulations, there can be exponentially many: we show a construction that has…

In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of {\em triangles}, for a matroid to have either $U_{2,4}$ or $F_7$ or $M(K_5)$ as a minor.

For a graph G, consider the pairs of edge-disjoint matchings whose union consists of as many edges as possible. Let H be the largest matching among such pairs. Let M be a maximum matching of G. We show that 5/4 is a tight upper bound for…

An elementary proof that certain pairs of $2\times 2$ matrices with nonnegative real coordinates generate free monoids.

In this note we show that the maximum number of vertices in any polyhedron $P=\{x\in \mathbb{R}^d : Ax\leq b\}$ with $0,1$-constraint matrix $A$ and a real vector $b$ is at most $d!$.

(I) We prove that the (maximum) number of monotone paths in a geometric triangulation of $n$ points in the plane is $O(1.7864^n)$. This improves an earlier upper bound of $O(1.8393^n)$; the current best lower bound is $\Omega(1.7003^n)$.…

We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz \cite{GK}, to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum…