AI 中文总结
本文针对r+1个顶点的r-均匀超图H_k^r，改进了其Turán密度π(H_3^r)的下界，并给出π(H_k^r)的统一误差界，得到多个渐近尖锐的结果。
AI 中文摘要
设π(H)为r-均匀超图H的Turán密度，H_k^r表示顶点数为r+1、边数恰好为k的r-均匀超图，其中1≤k≤r+1。Sidorenko（JCT-B，2024）证明，当r→∞时，π(H_3^r)≥(1.7215-o(1))r^{-2}，且对于固定的k，当r→∞时π(H_k^r)≥(C_k+o(1))r^{-(1+1/(k-2))}。Clemen随后将第一个界改进为：存在常数c>0使得π(H_3^r)≥cr^{-2}√(log r)。本文证明以下结果：1. 对任意固定ε>0，存在常数c_ε>0使得π(H_3^r)≥c_ε/(r(log r)^{2+ε})，结合已知上界π(H_3^r)≤1/r，可得π(H_3^r)=r^{-1+o(1)}；2. 对每个3≤k≤r+1，令s=min{k-2,r-k+2}，则0≤(k-2)/r - π(H_k^r)≤128/r(√(s log(er/s)) + log(er/s))，该估计给出π(H_k^r)的若干渐近尖锐结果，例如当log(er/k)=o(k)时，π(H_k^r)=(1+o(1))(k-2)/r。
英文摘要
Let $π(H)$ be the Turán density of an r-uniform hypergraph $H$ and let $H_k^r$ denote the $r$-uniform hypergraph on $r+1$ vertices with exactly $k$ edges, where $1\le k\le r+1$. Sidorenko~(JCT-B, 2024) proved that $π(H_3^r)\ge (1.7215-o(1))r^{-2}$ as $r\to\infty$ and $π(H_k^r)\ge (C_k+o(1))r^{-(1+1/(k-2))}$ for fixed $k$ as $r\to\infty$. Clemen~later improved the first bound to $π(H_3^r)\ge cr^{-2}\sqrt{\log r}$ for some constant $c>0$. In this article, we prove the following results. \begin{itemize} \item For any fixed $\varepsilon>0$, there is a constant $c_\varepsilon>0$ such that $$π(H_3^r)\ge \frac{c_\varepsilon}{r(\log r)^{2+\varepsilon}}.$$ %$π(H_3^r)\ge 1/(r(\log r)^{2+o(1)})$. Together with the known upper bound $π(H_3^r)\le1/r$, this implies $π(H_3^r)=r^{-1+o(1)}$. \item For every $3\le k\le r+1$, let $s=\min\{k-2,r-k+2\}$. Then \begin{equation*} 0\le \frac{k-2}{r}-π(H_k^r) \le \frac{128}{r}\left(\sqrt{s\log\frac{er}{s}}+\log\frac{er}{s}\right). \end{equation*} This estimate yields several asymptotically sharp results for $π(H_k^r)$. For example, $π(H_k^r)=(1+o(1))(k-2)/r$ when $\log(er/(k))=o(k)$. \end{itemize}
