arXiv subjects
Publications and source records attributed to Jim Geelen.
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.
We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids.
We prove, by means of an exact structural description, that every simple triangle-free binary matroid $M$ with $|M| > \tfrac{33}{128}2^{r(M)}$ has critical number at most $2$.
We prove that for each prime power $q$ there is an integer $n$ such that if $M$ is a $3$-connected, representable matroid with a PG$(n-1,q)$-minor and no $U_{2,q^2+1}$-minor, then $M$ is representable over GF$(q)$. We also show that for $\ell >= 2$, if $M$ is a $3$-connected, representable matroid of sufficiently high rank with no $U_{2,\ell+2}$-minor and $|E(M)| \geq (4\ell)^{r(M)/2}$, then $M$ is representable over a field of order at most $\ell$.
For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the structural characterization.
We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then there is a k-element subset X of E(M) such that one of M\X and M/X is 3-connected and contains N as a minor.
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 $d \le \tfrac{q^{2k}-1}{q-1}$ such that $h_{\mathcal{M}}(n) = \frac{q^{n+k}-1}{q-1} - qd$ for all sufficiently large $n$, and we characterise which matroids attain the growth rate function for large $n$. We also show that if $\mathcal{M}$ is specified in a certain `natural' way (by intersections of classes of matroids representable over different finite fields and/or by excluding a finite set of minors), then the constants $k$ and $d$, as well as the point that `sufficiently large' begins to apply to $n$, can be determined by a finite computation.
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.
For each odd integer $k\ge 5$, we prove that, if $M$ is a simple rank-$r$ binary matroid with no odd circuit of length less than $k$ and with $|M| > k 2^{r-k+1}$, then $M$ is isomorphic to a restriction of the rank-$r$ binary affine geometry; this bound is tight for all $r\ge k-1$. We use this to give a simpler proof of the following result of Govaerts and Storme: for each integer $n\ge 2$, if $M$ is a simple rank-$r$ binary matroid with no $PG(n-1,2)$-restriction and with $|M| > \left(1-\frac{11}{2^{n+2}}\right) 2^r$, then $M$ has critical number at most $n-1$. That result is a geometric analogue of a theorem of Andrasfai, Erdos, and Sos in extremal graph theory.
We show that, if $q$ is a prime power at most 5, then every rank-$r$ matroid with no $U_{2,q+2}$-minor has no more lines than a rank-$r$ projective geometry over GF$(q)$. We also give examples showing that for every other prime power this bound does not hold.
We prove that if M is a vertically 4-connected matroid with a modular flat X of rank at least three, then every representation of M | X over a finite field F extends to a unique F-representation of M. A corollary is that when F has order q, any vertically 4-connected matroid with a PG(2, F)-restriction is either F-representable or has a U_{2, q^2+1}-minor. We also show that no excluded minor for the class of F-representable matroids has a PG(2, F)-restriction.
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.
We show for each positive integer $a$ that, if $\cM$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $n$ such that either every rank-$r$ matroid in $\cM$ can be covered by at most $r^n$ sets of rank at most $a$, or $\cM$ contains the $\GF(q)$-representable matroids for some prime power $q$, and every rank-$r$ matroid in $\cM$ can be covered by at most $r^nq^r$ sets of rank at most $a$. This determines the maximum density of the matroids in $\cM$ up to a polynomial factor.
For a set $G$ of points in $\PG(m-1,q)$, let $\ex_q(G;n)$, denote the maximum size of a collection of points in $\PG(n-1,q)$ not containing a copy of $G$, up to projective equivalence. We show that \[\lim_{n\rightarrow \infty} \frac{\ex_q(G;n)}{|\PG(n-1,q)|} = 1-q^{1-c},\] where $c$ is the smallest integer such that there is a rank-$(m-c)$ flat in $\PG(m-1,q)$ that is disjoint from $G$. The result is an elementary application of the density version of the Hales-Jewett Theorem.
Let $\cM$ be a minor-closed class of matroids that does not contain arbitrarily long lines. The growth rate function, $h:\bN\rightarrow \bN$ of $\cM$ is given by $$h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$).$$ The Growth Rate Theorem shows that there is an integer $c$ such that either: $h(n)\le c\, n$, or ${n+1 \choose 2} \le h(n)\le c\, n^2$, or there is a prime-power $q$ such that $\frac{q^n-1}{q-1} \le h(n) \le c\, q^n$; this separates classes into those of linear density, quadratic density, and base-$q$ exponential density. For classes of base-$q$ exponential density that contain no $(q^2+1)$-point line, we prove that $h(n) =\frac{q^n-1}{q-1}$ for all sufficiently large $n$. We also prove that, for classes of base-$q$ exponential density that contain no $(q^2+q+1)$-point line, there exists $k\in\bN$ such that $h(n) = \frac{q^{n+k}-1}{q-1} - q\frac{q^{2k}-1}{q^2-1}$ for all sufficiently large $n$.
For any positive integer $l$ we prove that if $M$ is a simple matroid with no $(l+2)$-point line as a minor and with sufficiently large rank, then $|E(M)|\le \frac{q^{r(M)}-1}{q-1}$, where $q$ is the largest prime power less than or equal to $l$. Equality is attained by projective geometries over GF$(q)$.
It is proved that, for a prime number $p$, showing that an $n$-element matroid is not representable over $GF(p)$ requires only $O(n^2)$ rank evaluations.
It is proved that for each prime field $GF(p)$, there is an integer $f(p)$ such that a 4-connected matroid has at most $f(p)$ inequivalent representations over $GF(p)$. We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term "$k$-coherence". We obtain a variety of other results on inequivalent representations including the following curious one. For a prime power $q$, let ${\mathcal R}(q)$ denote the set of matroids representable over all fields with at least $q$ elements. Then there are infinitely many Mersenne primes if and only if, for each prime power $q$, there is an integer $m_q$ such that a 3-connected member of ${\mathcal R}(q)$ has at most $m_q$ inequivalent GF(7)-representations. The theorems on inequivalent representations of matroids are consequences of structural results that do not rely on representability. The bulk of this paper is devoted to proving such results.