Rational codegree Turán density of hypergraphs
Jun Gao, Oleg Pikhurko, Mingyuan Rong, Shumin Sun
TL;DR
The paper studies the codegree Turán density $\gamma(\mathcal{F})$ for $k$-graphs ($k\ge3$), proving that every rational density in $[0,1)$ can be realized by a finite forbidden family and that strong non-principality holds: there exist $F_1,F_2$ with $0<\gamma(\{F_1,F_2\})<\min\{\gamma(F_1),\gamma(F_2)\}$. The constructive backbone uses the $(s,t)$-extension and the $(a,b+1)$-extension, together with Mubayi– Zhao's lemma, to realize any rational density; the paper also builds a natural family $F^k_r$ with $\gamma(F^k_r)=(r-1)/r$. The non-principality result follows by combining two explicit graphs, $K^k(\ell,\ell)$ and $F^k_2$, to show a density gap at $1/2$ for their joint forbidden family, resolving a question of Mubayi and Zhao. These results deepen our understanding of the density spectrum in hypergraphs and extend density phenomena beyond the graph case.
Abstract
Let $H$ be a $k$-graph (i.e. a $k$-uniform hypergraph). Its minimum codegree $δ_{k-1}(H)$ is the largest integer $t$ such that every $(k-1)$-subset of $V(H)$ is contained in at least $t$ edges of~$H$. The \emph{codegree Turán density} $γ(\mathcal{F})$ of a family $\mathcal{F}$ of $k$-graphs is the infimum of $γ> 0$ such that every $k$-graph $H$ on $n\to\infty$ vertices with $δ_{k-1}(H) \ge (γ+o(1))\, n$ contains some member of $\mathcal{F}$ as a subgraph. We prove that, for every integer $k\ge3$ and every rational number $α\in [0,1)$, there exists a finite family of $k$-graphs $\mathcal{F}$ such that $γ(\mathcal{F})=α$. Also, for every $k \ge 3$, we establish a strong version of non-principality, namely that there are two $k$-graphs $F_1$ and $F_2$ such that the codegree Turán density of $\{F_1,F_2\}$ is strictly smaller than that of each $F_i$. This answers a question of Mubayi and Zhao [J Comb Theory (A) 114 (2007) 1118--1132].