We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some k, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with t triples spanning $ O(\sqrt t )$ vertices, which is the best possible up to the implied constant. We confirm that for all t we can find a collection of t triples spanning at most t + 3 vertices, resolving the Brown–Erdős–Sós conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemerédi’s theorem and the density Hales–Jewett theorem.This result was discovered simultaneously and independently by Nenadov, Sudakov and Tyomkyn [5], and a weaker result avoiding the arithmetic machinery was obtained independently by Wong [11].

中文翻译：

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关于群中 Brown-Erdős-Sós 猜想的注释

我们证明了一个足够大的群乘法表的密集子集包含以一些为模的整数加法表的大部分ķ，或者某个大阿贝尔群的整个乘法表，作为一个子网格。因此，我们证明了来自有限群的三元组系统包含具有吨三元组跨越$ O(\sqrt t )$顶点，这可能是隐含常数的最佳值。我们确认，对于所有人吨我们可以找到一个集合吨三元组最多跨越吨+ 3 个顶点，在这种情况下解决了 Brown-Erdős-Sós 猜想。该证明应用了著名的算术结果，包括 Szemerédi 定理和密度 Hales-Jewett 定理的多维版本。该结果由 Nenadov、Sudakov 和 Tyomkyn [5] 同时独立地发现，并获得了避开算术机制的较弱结果由 Wong [11] 独立完成。