A classical theorem of Simonovits from the 1980s asserts that every graph $G$ satisfying $e\left(G\right)\gg v{\left(G\right)}^{1+1∕k}$ must contain $\gtrsim {\left(\frac{e\left(G\right)}{v\left(G\right)}\right)}^{2k}$ copies of $C_{2k}$. Recently, Morris and Saxton established a balanced version of Simonovits' theorem, showing that such $G$ has $\gtrsim {\left(\frac{e\left(G\right)}{v\left(G\right)}\right)}^{2k}$ copies of $C_{2k}$, which are “uniformly distributed” over the edges of $G$. Moreover, they used this result to obtain a sharp bound on the number of $C_{2k}$-free graphs via the method of hypergraph containers. In this article, we generalise Morris–Saxton's results for even cycles to $\mathrm{\Theta }$-graphs. We also prove analogous results for complete $r$-partite $r$-graphs.

中文翻译：

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一些退化超图的平衡过饱和

1980 年代 Simonovits 的一个经典定理断言，每个图 $G$ 满意 $\mathrm{电子}\left(G\right)\gg v{\left(G\right)}^{1+1∕克}$ 必须包含 $\gtrsim {\left(\frac{\mathrm{电子}\left(G\right)}{v\left(G\right)}\right)}^{2克}$ 副本 $C_{2克}$. 最近，莫里斯和萨克斯顿建立了西蒙诺维茨定理的平衡版本，表明这样的$G$ 具有 $\gtrsim {\left(\frac{\mathrm{电子}\left(G\right)}{v\left(G\right)}\right)}^{2克}$ 副本 $C_{2克}$，它们“均匀分布”在 $G$. 此外，他们使用这个结果来获得对数量的急剧限制$C_{2克}$通过超图容器的方法的自由图。在本文中，我们将 Morris-Saxton 对偶数周期的结果概括为$\mathrm{\Theta }$-图表。我们还证明了完整的类似结果$r$-方 $r$-图表。