CanaDAM 2019
SFU Harbour Centre, 29 - 31 mai 2019

Matroid theory II
Org: Peter Nelson (University of Waterloo)

ANN-KATHRIN ELM, University of Hamburg
Infinite (pseudo-)flowers  [PDF]

Flowers are structures in matroids which display many separations of the same connectivity. There is a generalisation of a certain type of flower (anemones). We are going to explain why this generalisation is useful in infinite matroids, why there is no need to generalise other flowers, and give some properties of this generalisation.

DARYL FUNK, Douglas College
Describing quasi-graphic matroids  [PDF]

The class of quasi-graphic matroids recently introduced by Geelen, Gerards, and Whittle generalises each of the classes of frame and lifted-graphic matroids introduced earlier by Zaslavsky. We show that every quasi-graphic matroid has a representation as a graph together with a partition $(\mathcal B, \mathcal L, \mathcal F)$ of its cycles satisfying two natural conditions. Conversely, every graph equipped with such a partition defines a quasi-graphic matroid. From this follow cryptomorphic descriptions in terms of circuits, cocircuits, independent sets, and bases. These descriptions enable us to prove some results on quasi-graphic matroids.

Joint with Nathan Bowler and Daniel Slilaty.

JORN VAN DER POL, University of Waterloo
Enumerating matroids of fixed rank  [PDF]

We consider the number of matroids on an $n$-element ground set where the rank is fixed.

We obtain asymptotically precise upper and lower bounds on the number of matroids of rank at least 4; a gap remains for rank-3 matroids.

Of particular interest are sparse paving matroids. These well-behaved matroids are related to both Steiner systems and matchings in hypergraphs. Existing enumeration methods (Linial-Luria, Bennett-Bohman) that work in those settings can be made to work for sparse paving matroids as well.

A combinatorial argument then extends these bounds to the broader class of paving matroids, and thence to general matroids.

ZACHARY WALSH, University of Waterloo
Quadratically Dense Matroids  [PDF]

The Growth Rate Theorem of Geelen, Kabell, Kung, and Whittle says that the extremal function of a minor-closed class of matroids which excludes a line is either linear, quadratic, or exponential. We discuss work towards showing that Dowling geometries are essentially the densest matroids in quadratically dense minor-closed classes which exclude a doubled clique.