发表机构
University of Warwick; Shanghai Jiao Tong University; Hong Kong University of Science and Technology（华威大学; 上海交通大学; 香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究3-一致超图的Brown--Erdős--Sós问题，证明π(9)=1/5，并给出π(8)的上界5053/26544，通过线性规划与有限域构造实现。
AI 中文摘要
Brown--Erdős--Sós于1973年提出的著名且被积极研究的问题，询问$f^{(r)}(n;s,k)$，即在具有$n$个顶点的$r$-图中，不存在$s$个顶点张成$k$条或更多边时的最大边数。本文集中研究$r=3$且$s=k+2$的情形，其中$k\ge2$固定且$n\to\infty$；此时容易证明极值函数关于$n$二次增长。Delcourt和Postle证明了对于每个$k$，极限$\pi(k):=\lim_{n\to\infty} f^{(3)}(n;k+2,k)/n^2$存在。虽然Brown、Erdős和Sós在20世纪70年代已观察到$\pi(2)=1/6$，但$3\le k\le 7$时$\pi(k)$的值直到最近才被确定（由Glock、Joos、Kim、Kühn、Lichev、Pikhurko和Sun的不同子群体完成）。非常近期，Chao、Huang和Liu确定了所有奇数$k$的$\pi(k)$。独立于最后这一结果，我们证明$\pi(9)=1/5$。此外，我们证明$\pi(8)\le {5053}/{26544}$，这与已知最佳下界$\pi(8)\ge 3/16$相差在0.0029以内。新的上界通过将先前的一些论证表达为线性规划，然后使用计算机生成并求解其实例而获得。我们对$\pi(9)$下界的证明基于有限域构造并结合现有的打包结果。
英文摘要
The famous and actively studied problem of Brown--Erdős--Sós from 1973 asks for $f^{(r)}(n;s,k)$, the maximum number of edges in an $r$-graph with $n$ vertices in which no $s$ vertices span $k$ or more edges. In this paper, we concentrate on the case $r=3$ and $s=k+2$, with $k\ge2$ fixed and $n\to\infty$; then it is easy to show that the extremal function grows quadratically in $n$. Delcourt and Postle proved that the limit $π(k):=\lim_{n\to\infty} f^{(3)}(n;k+2,k)/n^2$ exists for every $k$. While Brown, Erdős and Sós observed that $π(2)=1/6$ already in the 1970s, the value of $π(k)$ for $3\le k\le 7$ was determined only recently (by various subgroups of Glock, Joos, Kim, Kühn, Lichev, Pikhurko, and Sun). Very recently, Chao, Huang and Liu determined $π(k)$ for every odd $k$. Independently of the last result, we show that $π(9)=1/5$. Also, we prove that $π(8)\le {5053}/{26544}$, which is within $0.0029$ of the best known lower bound $π(8)\ge 3/16$. The new upper bounds are obtained by expressing some previous arguments as a linear program and then using a computer to generate and solve its instances. Our proof of the lower bound on $π(9)$ is based on a finite field construction combined with existing packing results.
