Common linear patterns are rare

Abstract:

A linear system $L$ over $\mathbb{F}_q$ is common if the number of monochromatic solutions to $L=0$ 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. Fox, Pham and Zhao characterised common linear equations.

I will talk about recent progress towards a classification of common systems of two or more linear equations. In particular, any system containing a four-term arithmetic progression is uncommon. 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.

Joint work with Anita Liebenau and Natasha Morrison.

A linear system LL over FqFq is common if the number of monochromatic solutions to L=0L=0 in any two-colouring of FnqFqn is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of FnqFqn. 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. Fox, Pham and Zhao characterised common linear equations.

I will talk about recent progress towards a classification of common systems of two or more linear equations. In particular, any system containing a four-term arithmetic progression is uncommon. 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.

Joint work with Anita Liebenau and Natasha Morrison.

This is a seminar of the Combinatorial Mathematics Society of Australasia.

Zoom link: https://unsw.zoom.us/j/99740631801?pwd=c09qVTkwUGoxMGJyQjFsSmtmamd3UT09

Meeting ID: 997 4063 1801

Passcode: 135082