Uncommon linear systems of two equations

TL;DR

This paper classifies when certain small systems of two linear equations over finite fields are common or uncommon, extending previous work and answering open questions in the field.

Contribution

It determines the commonness of all 2x5 linear systems except for a few cases and proves that all 2xk systems with even k and girth k-1 are uncommon.

Findings

Complete classification of 2x5 systems' commonness.

All 2xk systems with even k and girth k-1 are uncommon.

Answers to open questions about the structure of common linear systems.

Abstract

A system of linear equations $L$ is common over $\mathbb{F}_p$ if, as $n\to\infty$, any 2-coloring of $\mathbb{F}_p^n$ gives asymptotically at least as many monochromatic solutions to $L$ as a random 2-coloring. The notion of common linear systems is analogous to that of common graphs, i.e., graphs whose monochromatic density in 2-edge-coloring of cliques is asymptotically minimized by the random coloring. Saad and Wolf initiated a systematic study on identifying common linear systems, built upon the earlier work of Cameron-Cilleruelo-Serra. When $L$ is a single equation, Fox-Pham-Zhao gave a complete characterization of common linear equations. When $L$ consists of two equations, Kam\v{c}ev-Liebenau-Morrison showed that irredundant $2\times 4$ linear systems are always uncommon. In this work, (1) we determine commonness of all $2\times 5$ linear systems up to a small number of cases, and (2) we show that all $2\times k$ linear systems with $k$ even and girth (minimum number of nonzero coefficients of a nonzero equation spanned by the system) $k-1$ are uncommon, answering a question of Kam\v{c}ev-Liebenau-Morrison.