Linear configurations containing 4-term arithmetic progressions are uncommon

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 Fp for p ≥ 5 and large n and in Zp for large primes p.

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