A result due to Gyárfás, Hubenko, and Solymosi (answering a question of Erdős) states that if a graph G on n vertices does not contain K 2,2 as an induced subgraph yet has at least $$c\left(\begin{array}{c}n\\ 2\end{array}\right)$$ c ( n 2 ) edges, then G has a complete subgraph on at least $$\frac{c^2}{10}n$$ c 2 10 n vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced K 2,2 which allows us to generalize their result to k -uniform hypergraphs. Our result also has an interesting consequence in discrete geometry. In particular, it implies that the fractional Helly theorem can be derived as a purely combinatorial consequence of the colorful Helly theorem .

中文翻译：

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具有禁止子结构的超图中的大集团

Gyárfás、Hubenko 和 Solymosi 的结果（回答 Erdős 的问题）指出，如果 n 个顶点上的图 G 不包含 K 2,2 作为诱导子图，但至少有 $$c\left(\begin{ array}{c}n\\ 2\end{array}\right)$$ c ( n 2 ) 个边，那么 G 至少在 $$\frac{c^2}{10}n$$ 上有一个完整的子图c 2 10 n 个顶点。在本文中，我们提出了诱导 K 2,2 概念的“高维”类似物，这使我们能够将他们的结果推广到 k 均匀超图。我们的结果在离散几何中也有一个有趣的结果。特别是，它意味着分数 Helly 定理可以作为彩色 Helly 定理的纯组合结果推导出来。