AI 中文总结
本文给出完全\(r\)部\(r\)均匀超图图兰数的埃尔德什经典定理的另一种证明。通过多次应用赫尔德不等式及多重求和枚举构型，结合容斥原理与富比尼定理离散类似物，得出递归估计，为埃尔德什经典上界提供新证并统一视角。
AI 中文摘要
在本文中，我们给出了关于完全\(r\)部\(r\)均匀超图的图兰数的埃尔德什经典定理的另一种证明。具体而言，对于有限集\(A_{1},\ldots,A_{r}\)，其中\(|A_{1}|\leq\cdots\leq|A_{r}|\)以及足够大的正整数\(n\)，我们证明了\(\mathrm{ex}(n,\mathbb{K}^{(r)}[A_{1},\ldots,A_{r}]) =O\left(n^{r-\frac{1}{|A_{k}|\cdots|A_{r-1}|}}\right)\)。我们的方法基于多次应用赫尔德不等式以及通过多重求和对构型进行枚举来构建框架。该方法将容斥原理与富比尼定理的离散类似物相结合，得出极值量的递归估计。这为埃尔德什的经典上界提供了另一种证明，并为广义埃尔德什盒问题提供了统一视角。
英文摘要
In this article, we develop an $r-$uniform analogue of the classical Kövári--Sós--Turán inequality. This yields an alternative proof of Erdős's classical upper bound for complete $r$-partite $r$-uniform hypergraphs. More precisely, we establish that for finite non-empty sets $A_{1},\ldots,A_{r}$ with $|A_{1}|\leq\cdots\leq|A_{r}|$ and sufficiently large positive integer $n$, \[\mathrm{ex}(n,\mathbb{K}^{(r)}[A_{1},\ldots,A_{r}])=O\left(n^{r-\frac{1}{|A_{1}|\ldots|A_{r-1}|}}\right).\] Our proof develops an analytic counting framework based on repeated applications of Hölder's inequality and the enumeration of configurations through multiple sums. The argument combines the principle of inclusion--exclusion with a discrete analogue of Fubini's theorem to obtain recursive estimates for extremal quantities. This provides a unified analytic perspective on the generalized Erdős box problem.
Comments15 pages
