Matroid theory
Organizer and Chair: Stefan van Zwam (Louisiana State University, USA)
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JOSEPH BONIN, The George Washington University
Excluded Minors for (Strongly) Base-Orderable Matroids  [PDF]

The minor-closed classes of base-orderable matroids and strongly base-orderable matroids properly include transversal matroids and gammoids. In 1976, Ingleton sketched how one could, in theory, find the infinite set of excluded minors for base-orderable matroids; he gave examples, attractive conjectures, and results (without proofs). We approach Ingleton's ideas from the perspective of cyclic flats and make progress on his conjectures. Ingleton gave an example of a matroid that is base-orderable but not strongly base-orderable; building on that, we show that the class of strongly base-orderable matroids has infinitely many base-orderable excluded-minors.

This is joint work with Thomas Savitsky.

DILLON MAYHEW, Victoria University of Wellington
Fans and fragile matroids  [PDF]

Let $\mathcal{S}$ be a set of matroids. The matroid $M$ is $\mathcal{S}$-fragile if, for every element $e$ in the ground set of $M$, either $M\backslash e$ or $M/e$ has no minor isomorphic to a member of $\mathcal{S}$. Fragile classes are important in excluded-minor proofs.

When characterising fragile classes, it seems sometimes the only way we can build from a fragile matroid while remaining fragile is to grow fans. How can we know this is the case? We have proved a theorem that reduces this question to a finite-case analysis.

Joint work with Carolyn Chun, Deb Chun, and Stefan van Zwam.

PETER NELSON, University of Waterloo
Matroids denser than a projective geometry  [PDF]

The unique densest simple rank-$n$ matroid representable over a finite field GF$(q)$ is the projective geometry over GF$(q)$, which has $\frac{q^n-1}{q-1}$ elements. Moreover, a theorem of Kung shows that any denser matroid has a $U_{2,q+2}$-minor , which is a simple certificate of non-GF$(q)$-representability. I will discuss a theorem that shows that much richer minors can be found in every large-rank matroid denser than a projective geometry over GF$(q)$. I will also state many corollaries that bound the growth rate function of most classes obtained by excluding free spikes, free swirls and/or lines as minors.

IRENE PIVOTTO, The University of Western Australia
A first step in decomposing near-regular matroids  [PDF]

The prototypical example of a matroid decomposition theorem is Seymour's result that any regular matroid may be constructed from graphic and cographic matroids and copies of $R_{10}$ via 1-, 2- and 3-sums. Whittle conjectured that a similar decomposition theorem holds for the more general class of near-regular matroids. This conjecture, later amended by Mayhew, Whittle and van Zwam to include 4-sums, is still open. In this talk we present a first step toward resolving this conjecture, namely a decomposition theorem for single element extensions and coextensions of graphic matroids. This is joint work with D. Chun and D. Slilaty.

STEFAN VAN ZWAM, Louisiana State University
Minor-closed classes have no asymptotically good codes  [PDF]

A binary, linear, $[n,r]$ error-correcting code is an $r$-dimensional subspace of $\textrm{GF}(2)^n$. The distance between two codewords is the number of coordinates where they differ, and the minimum distance is denoted by $d_C$. A family of codes is asymptotically good if there exists an $\epsilon>0$ and codes in the family with $r_C/n_C\geq\epsilon$, $d_C/n_C\geq\epsilon$, and $n_C$ arbitrarily large.

We show that structured families are not asymptotically good. In this case, “structured” means that the corresponding matroids form a proper minor-closed class. Our proof is built on the Matroid Structure Theorem by Geelen, Gerards, and Whittle.

Joint work with P. Nelson