Monday, July 19, 3pm ET (8pm BST, 7am Tue NZST)
Natasha Morrison, University of Victoria
Uncommon systems of equations

Abstract:

A system of linear equations $L$ over $\mathbb{F}_q$ is common if the number of monochromatic solutions to $L$ in any two-colouring of $\mathbb{F}_q^n$ is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of $\mathbb{F}_q^n$. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Building on earlier work of Cameron, Cilleruelo and Serra, as well as Saad and Wolf, common linear equations have been fully characterised by Fox, Pham and Zhao.

In this talk I will discuss some recent progress towards a characterisation of common systems of two or more equations. In particular we prove that any system containing an arithmetic progression of length four is uncommon, confirming a conjecture of Saad and Wolf. This follows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsystems.

This is joint work with Nina Kamčev and Anita Liebenau.

Monday, July 12, 3pm ET (8pm BST, 7am Tue NZST)
Eimear Byrne, University College Dublin
Some recent results on q-matroids

Abstract:
The q-analogue of a (poly)matroid has been a topic of recent interest among the coding theory community, due to their connections with rank-metric codes (see e.g. [GJLR20], [JP18], [Shi19]). When defining a q-(poly)matroid, we associate a rank function with the lattice of subspaces of a finite dimensional space E. As one might expect, there are several cryptomorphic descriptions of a q-matroid, in terms of independent spaces, circuits, flats, etc. We will go through some of these, highlighting the difference to the classical case. We will also outline some applications of q-(poly)matroids to the construction of the q-analogue of a t-design.

Some of the results of this talk are based on joint work with Michela Ceria, Sorina Ionica, Relinde Jurrius and Elif Sacikara.

[JP18] Relinde Jurrius and Ruud Pellikaan. Defining the q-analogue of a matroid. Electronic Journal of Combinatorics, 25(3):P3.2, 2018.

[GJLR20] Elisa Gorla, Relinde Jurrius, Hiram H L´opez, and Alberto Ravagnani. Rank-metric codes and q-polymatroids. Journal of Algebraic Combinatorics, 52:1–19, 2020.

[Shi19] Keisuke Shiromoto. Codes with the rank metric and matroids. Designs, Codes and Cryptography, 87(8):1765–1776, 2019.

Monday, July 5, 3pm ET (8pm BST, 7am Tue NZST)
Shayla Redlin, University of Waterloo
Extensions of cliques

Abstract:

In this talk, we will investigate the extensions of the cycle matroid of a complete graph. I will show that the number of these extensions is surprisingly large.

This is joint work with Peter Nelson and Jorn van der Pol.

Monday, June 28, 3pm ET (8pm BST, 7am Tue NZST)
Carla Groenland, Utrecht University
Universal Graphs and Labelling Schemes

Abstract:
An induced universal graph for a graph class contains all graphs in the class as an induced subgraph. We construct induced universal graphs for all hereditary graph classes, and derive reachability labelling schemes for digraphs and comparability labelling schemes for posets from this. All these results are asymptotically optimal. This talk aims to give some intuition about these concepts and our techniques (which includes Szemerédi’s regularity lemma). We will also discuss a new venue of research: what if we strengthen induced to isometric? Several interesting questions are left open.

This is based on joint work with Marthe Bonamy, Louis Esperet, Cyril Gavoille and Alex Scott.