Near-optimal Tur\'an densities of $r$-graphs on $r+1$ vertices

Abstract

Let $\pi(H)$ be the Tur\'an 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 $\pi(H_3^r)\ge (1.7215-o(1))r^{-2}$ as $r\to\infty$ and $\pi(H_k^r)\ge (C_k+o(1))r^{-(1+1/(k-2))}$ for fixed $k$ as $r\to\infty$. Clemen~later improved the first bound to $\pi(H_3^r)\ge cr^{-2}\sqrt{\log r}$ 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 $$\pi(H_3^r)\ge \frac{c_\varepsilon}{r(\log r)^{2+\varepsilon}}.$$ %$\pi(H_3^r)\ge 1/(r(\log r)^{2+o(1)})$. Together with the known upper bound $\pi(H_3^r)\le1/r$, this implies $\pi(H_3^r)=r^{-1+o(1)}$. \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}-\pi(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 $\pi(H_k^r)$. For example, $\pi(H_k^r)=(1+o(1))(k-2)/r$ when $\log(er/(k))=o(k)$. \end{itemize}