Linear configurations containing 4-term arithmetic progressions are uncommon

TL;DR

This paper proves that linear configurations containing 4-term arithmetic progressions are generally uncommon in large finite fields and cyclic groups, extending previous graph theory results to algebraic structures.

Contribution

It establishes that all such configurations are indeed uncommon in certain algebraic settings, answering a question posed by Saad and Wolf.

Findings

Linear configurations with 4-term arithmetic progressions are uncommon in large finite fields.

The result holds for $ ext{F}_p^n$ with $p extgreater 4$ and large $n$.

The result also applies to cyclic groups $ ext{Z}_p$ for large primes $p$.

Abstract

A linear configuration is said to be common in $G$ if every 2-coloring of $G$ yields at least the number of monochromatic instances of a randomly chosen coloring. Saad and Wolf asked whether, analogously to a result by Thomason in graph theory, every configuration containing a 4-term arithmetic progression is uncommon. We prove this in $\mathbb{F}_p^n$ for $p\geq 5$ and large $n$ and in $\mathbb{Z}_p$ for large primes $p$.