发表机构
Tel Aviv University; University of Haifa（特拉维夫大学; 海法大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究解决了Erdős–Stone问题在边密度接近1时的界的紧性问题，确定了该情形下$b_n(r,δ)$的正确阶，给出了适用于所有边密度的紧界，并分别通过构造性方法证明了上下界。
AI 中文摘要
Erdős–Stone定理断言，若一个图的边密度为$1-1/r+δ$，则它包含一个各部分有$b$个顶点的完全$(r+1)$部图，其中$b=b_n(r,δ) \gg 1$。著名的Chvátal–Szemerédi定理确定了对所有$δ< 1/r^3$，$b_n(r,δ)$的精确阶。然而，当$δ=1/r-ε$时，即当图的边密度为$1-ε$且$ε$较小时，他们的界并不紧。本文的主要结果确定了这一剩余情形下的正确阶，从而使我们能够给出Erdős–Stone问题对所有边密度的紧界。更确切地说，我们证明对每个整数$r\geq 2$和$0< δ< 1/r$，有$$ b_n(r,δ)=Θ\left(\frac{\log n}{(1/r-δ)r\log(1/δ)}\right)\\\\;. $$下界通过Kövari-Sós-Turán型论证结合Nikiforov构造大blow-up的方法的一个变体得到，而上界则利用与张量幂相关的相关随机图构造证明。
英文摘要
The Erdős--Stone Theorem asserts that if a graph has edge density $1-1/r+δ$ then it contains a complete $(r+1)$-partite graph with $b$ vertices in each part, where $b=b_n(r,δ) \gg 1$. The celebrated Chvátal--Szemerédi theorem determined the exact order of $b_n(r,δ)$ for every $δ< 1/r^3$. Their bound, however, is not tight when $δ=1/r-ε$, that is, when the graph has edge density $1-ε$ for small $ε$. Our main result in this paper determines the correct order in this remaining regime, thereby enabling us to give a tight bound for the Erdős--Stone problem for all edge densities. More precisely, we prove that for every integer $r\geq 2$ and $0< δ< 1/r$ we have $$ b_n(r,δ)=Θ\left(\frac{\log n}{(1/r-δ)r\log(1/δ)}\right)\;. $$ The lower bound is obtained using a Kövari-Sós-Turán-type argument combined with a variant of Nikiforov's method of constructing large blow-ups, while the upper bound is proved using a correlated random graph construction, related to tensor powers.
