Fix $k \geq 6$. We prove that any large enough finite group $G$ contains $k$ elements which span quadratically many triples of the form $(a,b,ab) \in S \times G$, given any dense set $S \subseteq G \times G$. The quadratic bound is asymptotically optimal. In particular, this provides an elementary proof of a special case of a conjecture of Brown, Erd\H{o}s and S\'{o}s. We remark that the result was recently discovered independently by Nenadov, Sudakov and Tyomkyn.

中文翻译：

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关于有限群中稠密子结构的存在性

修复 $k \geq 6$。我们证明任何足够大的有限群 $G$ 包含 $k$ 个元素，这些元素跨越形式为 $(a,b,ab) \in S \times G$ 的二次三元组，给定任何稠密集 $S \subseteq G \倍 G$。二次界是渐近最优的。特别是，这提供了 Brown、Erd\H{o}s 和 S\'{o}s 猜想的特殊情况的基本证明。我们注意到该结果是最近由 Nenadov、Sudakov 和 Tyomkyn 独立发现的。