On uncommon systems of equations

TL;DR

This paper investigates when systems of linear equations over finite fields are common, proving that systems containing a four-term arithmetic progression are uncommon, advancing the classification of such systems.

Contribution

It extends the classification of common systems of linear equations by proving that systems with a four-term arithmetic progression are necessarily uncommon.

Findings

Systems with a four-term arithmetic progression are uncommon.

Uncommonness of a system can be deduced from properties of its subsystems.

The work advances the understanding of the structure of common and uncommon systems.

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 upon earlier work of Cameron, Cilleruelo and Serra, as well as Saad and Wolf, common linear equations have recently been fully characterised by Fox, Pham and Zhao, who asked about common \emph{systems} of equations. In this paper we move towards a classification of common systems of two or more linear 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.