Rational codegree Turán density of hypergraphs

Abstract: Let $H$ be a $k$-graph (i.e. a $k$-uniform hypergraph). Its minimum codegree $δ<em>{k-1}(H)$ 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 $γ&gt; 0$ such that every $k$-graph $H$ on $n\to\infty$ vertices with $δ</em>{k-1}(H) \ge (γ+o(1))\, n$ 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].