Turán densities of stars in uniformly dense hypergraphs
Hao Lin, Wenling Zhou
TL;DR
The paper addresses the problem of determining the dot and dot-edge uniform Turán densities of the $S_k$-star in 3-uniform hypergraphs under $(d,\mu)$-dense constraints. It adopts the palette framework of Lamaison and Ander to reduce the problem to bounding densities of $S_k$-bad palettes and proves a Tk-free-digraph Caro–Wei–type bound to control the palette data. The main contributions are the proof that $π_{dot}(S_k)=(k^2-5k+7)/(k-1)^2$ for all $k\ge 11$ and the determination of $π_{dot-edge}(S_k)$ for all $S_k$ with $k\neq 4$ (avoiding hypergraph regularity). These results advance the understanding of uniform Turán densities for stars and demonstrate the effectiveness of palette-based methods in dense hypergraph extremal problems.
Abstract
A $3$-uniform hypergraph (or $3$-graph) $H=(V,E)$ is $(d,μ, \text{dot})$-dense if for any subsets $X, Y, Z\subseteq V$, the number of triples $(x,y,z)\in X\times Y\times Y$ with $\{x,y,z\}$ being an edge of $H$ is at least $d|X||Y||Z|-μ|V|^3$. Similarly, we say that $H$ is $(d,μ, \text{dot-edge})$-dense if for any subset $X\subseteq V$ and every pair set $P\subseteq V\times V$, the number of pairs $(x,(y,z))\in X\times P$ with $\{x,y,z\}$ being an edge of $H$ is at least $d|X||P|-μ|V|^3$. Restricting to $\text{dot}$-dense $3$-graphs and $\text{dot-edge}$-dense $3$-graphs, determining the $\text{dot}$-uniform Turán density $π_{\text{dot}}(S_k)$ and the $\text{dot-edge}$-uniform Turán density $π_{\text{dot-edge}}(S_k)$ of the $k$-star $S_k$ for $k\ge 4$ was proposed by Schacht in ICM 2022. In particular, Reiher, Rödl and Schacht presented that $π_{\text{dot}}(S_k)\ge π_{\text{dot-edge}}(S_k)\ge \frac{k^2-5k+7}{(k-1)^2}$ for $k\ge 3$ and $π_{\text{dot}}(S_3)= π_{\text{dot-edge}}(S_3)=1/4$. Last year, Lamaison and Wu shown that $π_{\text{dot}}(S_k)=\frac{k^2-5k+7}{(k-1)^2}$ for $k\ge 48$. In this paper, we show that $π_{\text{dot}}(S_k)=\frac{k^2-5k+7}{(k-1)^2}$ for $k\ge 11$. Moreover, we determine the $\text{dot-edge}$-uniform Turán density for all $S_k$ except for $k=4$.