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, Šťovíček and Thomason in graph theory, every configuration containing a 4-term arithmetic progression is uncommon. We prove this in ${\mathbb{F}}_p^n$ for $p\ge 5$ and large n and in ${\mathbb{Z}}_p$ for large primes p.

中文翻译：

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包含 4 项算术级数的线性配置并不常见

如果G的每个 2 着色至少产生随机选择的着色的单色实例的数量，则称线性配置在阿贝尔群G中是常见的。Saad 和 Wolf 询问，是否与 Jagger、Šťovíček 和 Thomason 在图论中的结果类似，包含 4 项算术级数的每个配置都是不常见的。我们证明了这一点${\mathbb{F}}_p^n$为了$p\ge 5$和大n和 in${\mathbb{Z}}_p$对于大素数p。