Rational codegree Tur\'an density of hypergraphs

Let $H$ be a $k$-graph (i.e. a $k$-uniform hypergraph). Its minimum codegree $\delta_{k-1}(H)$ 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 Tur\'an density} $\gamma(\mathcal{F})$ of a family $\mathcal{F}$ of $k$-graphs is the infimum of $\gamma > 0$ such that every $k$-graph $H$ on $n\to\infty$ vertices with $\delta_{k-1}(H) \ge (\gamma+o(1))\, n$ contains some member of $\mathcal{F}$ as a subgraph. We prove that, for every integer $k\ge3$ and every rational number $\alpha \in [0,1)$, there exists a finite family of $k$-graphs $\mathcal{F}$ such that $\gamma(\mathcal{F})=\alpha$. 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 Tur\'an 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].