Hypergraphs with arbitrarily small codegree Turán density

Abstract: Let $k\geq 3$. 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 $\delta(H)\geq (\gamma + o(1))n$ contains a copy of $F$. As in 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.

We show that for every~$\eps>0$, there is a $k$-uniform hypergraph $F$ with $0<\gamma(F)<\eps$. This is in contrast to the classical Turán density, which cannot take any value in the interval $(0,k!/k^k)$ due to a fundamental result by Erd\H os.

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