Linear configurations containing 4-term arithmetic progressions are uncommon

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 $p$.