Hypergraphs with arbitrarily small codegree Turán density

Let $k\ge 3$. Given a $k$-uniform hypergraph $H$, the minimum codegree $\delta (H)$ is the largest $d\in \mathbb{N}$ such that every $(k-1)$-set of $V(H)$ is contained in at least $d$ edges. Given a $k$-uniform hypergraph $F$, the codegree Turán density $\gamma (F)$ of $F$ is the smallest $\gamma \in [0,1]$ such that every $k$-uniform hypergraph on $n$ vertices with $\delta (H)\ge (\gamma +o(1))n$ contains a copy of $F$. Similarly as other variants of the hypergraph Turán problem, determining the codegree Turán density of a hypergraph is in general notoriously difficult and only few results are known. In this work, we show that for every $\epsilon >0$, there is a $k$-uniform hypergraph $F$ with $0<\gamma (F)<\epsilon$. This is in contrast to the classical Turán density, which cannot take any value in the interval $(0,k!\mathrm{/}{k}^k)$ due to a fundamental result by Erdős.