Linear configurations containing 4-term arithmetic progressions are uncommon

A linear configuration is said to be common in an Abelian group 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 Jagger, Stovi & ccaron;ek and Thomason in graph theory, every configuration containing a 4-term arithmetic progression is uncommon. We prove this in Fpn for p >= 5 and large n and in Zp for large primes p. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).